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Line 1: {{~~draft~~ task\|Probability and statistics~~}}{{draft task\|Discrete math~~}} {{wikipedia\|Combination~~}} {{wikipedia\|Permutation~~}} ~~Implement the [[wp:Combination\|combination]] (<sup>n</sup>C<sub>k</sub>) and [[wp:Permutation\|permutation]] (<sup>n</sup>P<sub>k</sub>) operators in the target language:~~ {{wikipedia\|Permutation}} * <math> ^nC_k =\binom nk = \frac{n(n-1)\ldots(n-k+1)}{k(k-1)\dots1} </math> * <math>^nP_k = n\cdot(n-1)\cdot(n-2)\cdots(n-k+1)</math> ~~See the wikipedia articles for a more detailed description.~~ ;Task: ~~To test, generate and print examples of:~~ Implement the [[wp:Combination\|combination]]   <big> (<sup>n</sup>C<sub>k</sub>) </big>   and [[wp:Permutation\|permutation]]   <big> (<sup>n</sup>P<sub>k</sub>) </big>   operators in the target language: * Permutations from 1 to 12 and Combinations from 10 to 60 using exact Integer arithmetic. * Permutations from 5 to 15000 and Combinations from 100 to 1000 using approximate Floating point arithmetic. :::* <math>^n\operatorname C_k =\binom nk = \frac{n(n-1)\ldots(n-k+1)}{k(k-1)\dots1} </math> ~~This 'floating point' code could be implemented using an approximation, e.g., by calling the [[Gamma function]].~~ :::* <math>^n\operatorname P_k = n\cdot(n-1)\cdot(n-2)\cdots(n-k+1)</math> <br> See the Wikipedia articles for a more detailed description. '''To test''', generate and print examples of: *   A sample of permutations from 1 to 12 and Combinations from 10 to 60 using exact Integer arithmetic. *   A sample of permutations from 5 to 15000 and Combinations from 100 to 1000 using approximate Floating point arithmetic.<br> This 'floating point' code could be implemented using an approximation, e.g., by calling the [[Gamma function]]. ;Related task: *   [[Evaluate binomial coefficients]] {{Template:Combinations and permutations}} <br><br> =={{header\|11l}}== {{trans\|C++}} <syntaxhighlight lang="11l">F perm(=n, p) BigInt r = 1 V k = n - p L n > k r = n-- R r F comb(n, =k) V r = perm(n, k) L k > 0 r I/= k-- R r L(i) 1..11 print(‘P(12,’i‘) = ’perm(12, i)) L(i) (10.<60).step(10) print(‘C(60,’i‘) = ’comb(60, i))</syntaxhighlight> {{out}} <pre> P(12,1) = 12 P(12,2) = 132 P(12,3) = 1320 P(12,4) = 11880 P(12,5) = 95040 P(12,6) = 665280 P(12,7) = 3991680 P(12,8) = 19958400 P(12,9) = 79833600 P(12,10) = 239500800 P(12,11) = 479001600 C(60,10) = 75394027566 C(60,20) = 4191844505805495 C(60,30) = 118264581564861424 C(60,40) = 4191844505805495 C(60,50) = 75394027566 </pre> =={{header\|Ada}}== Ada 2022 has Big_Integers, also we would probably use generic functions - but that's hard to show in a single file. <syntaxhighlight lang="ada"> pragma Ada_2022; with Ada.Text_IO; use Ada.Text_IO; with Ada.Numerics.Big_Numbers.Big_Integers; use Ada.Numerics.Big_Numbers.Big_Integers; with Interfaces; use Interfaces; procedure Combs_Perms is function U64_Perm (N, K : Unsigned_64) return Unsigned_64 is P : Unsigned_64 := 1; begin if K = 0 then P := 0; else for I in 0 .. K - 1 loop P := P (N - I); end loop; end if; return P; end U64_Perm; function Big_Perm (N, K : Natural) return Big_Natural is P : Big_Natural := 1; begin if K = 0 then P := 0; else for I in 0 .. K - 1 loop P := P * To_Big_Integer (N - I); end loop; end if; return P; end Big_Perm; function U64_Comb (N, K : Unsigned_64) return Unsigned_64 is Adj_K : constant Unsigned_64 := (if N - K < K then N - K else K); C : Unsigned_64 := U64_Perm (N, Adj_K); begin if K = 0 then C := 0; else for I in reverse 1 .. Adj_K loop C := C / I; end loop; end if; return C; end U64_Comb; function Big_Comb (N, K : Natural) return Big_Natural is Adj_K : constant Natural := (if N - K < K then N - K else K); C : Big_Natural := Big_Perm (N, Adj_K); begin for I in reverse 1 .. Adj_K loop C := C / To_Big_Integer (I); end loop; return C; end Big_Comb; begin Put_Line ("P(1, 0) =" & U64_Perm (1, 0)'Image); Put_Line ("P(12, 4) =" & U64_Perm (12, 4)'Image); Put_Line ("P(60, 20) =" & U64_Perm (60, 20)'Image); Put_Line ("P(105, 103) =" & Big_Perm (105, 103)'Image); Put_Line ("P(15000, 333) =" & Big_Perm (10000, 333)'Image); Put_Line ("C(10, 5) =" & U64_Comb (10, 5)'Image); Put_Line ("C(60, 30) =" & Big_Comb (60, 30)'Image); Put_Line ("C(900, 674) =" & Big_Comb (900, 674)'Image); end Combs_Perms; </syntaxhighlight> {{out}} <pre> P(1, 0) = 0 P(12, 4) = 11880 P(60, 20) = 1808792028959211520 P(105, 103) = 540698379120145450252050652900164824860323053887451289572088318286613265954952576663492268263404120169888199467436014828996936453906718408048640000000000000000000000000 P(15000, 333) = 3734632779825648626849478204217851175491031852573933200482258207588048438187890916059251484636506271051265834404767122580751173394985841384750846042214481658210687940637400976395676712758506534427611745860428779464238199671888490205087478284041341740481214200754003412080717927644948303742303217778756499322250874934482078713617010428749333465186089669624344010164333863671606638752312981517247538583321540255309837467575709099947985095703433121711204915009751758378839557821815211950252111673682998724830733294759824988032874155681853213162745288956061564026603816655765483177328085099709270712978486637711013862688274863138979103405877374342534841337812225037918093228818727609066843343063448062504973631670561421262499196336524811809011228722269099771902324209604494466003586741308023495564221843755524784021722020833255027012307883615611523716891547845775441485354168687580879756175578658189002755076470466059881166684817696505920834137318475281619250730364346027906505163737946409176743772248798012981461588038416291189183602392530371700797564272591564324880128780642945107817817148785181713276188934924102752418421382343547688708913308696047351312049027540308819760102775940016773345719986101919685659117157000736526082591575378358047866880000000000000000000000000000000000000000000000000000000000000000000000000000000000000 C(10, 5) = 252 C(60, 30) = 118264581564861424 C(900, 674) = 574717622813743458961613396879207645492072766667532001303398319071426357199738020963741184130318716754384770618027834663186286577340376549182421952347640973972155470158739236146281280772731378323732881115811005924314000 </pre> =={{header\|ALGOL 68}}== Line 15 ⟶ 155: {{works with\|ALGOL 68G\|Any - tested with release [http://sourceforge.net/projects/algol68/files/algol68g/algol68g-2.6 algol68g-2.6].}} {{wont work with\|ELLA ALGOL 68\|Any (with appropriate job cards) - tested with release [http://sourceforge.net/projects/algol68/files/algol68toc/algol68toc-1.8.8d/algol68toc-1.8-8d.fc9.i386.rpm/download 1.8-8d] - due to extensive use of '''format'''[ted] ''transput''.}} '''File: prelude_combinations_and_permutations.a68'''<~~lang~~syntaxhighlight lang="algol68"># -- coding: utf-8 -- # COMMENT REQUIRED by "prelude_combinations_and_permutations.a68" CO Line 36 ⟶ 176: IF n < r ORF r < 0 THEN IF NOT cp fix value error(CPARGS("P",n,r)) THEN stop FI FI; CPOUT out := 1; # basically nPk = (n-r+1)(n-r+2)...(n-2)(n-1)n = n!/(n-r+1)! # FOR i FROM n-r+1 TO n DO out := i OD; out Line 44 ⟶ 184: exp(ln gamma(n+1)-ln gamma(n-r+1)); # basically nPk = (n-r+1)(n-r+2)...(n-2)(n-1)n = n!/(n-r+1)! # COMMENT # alternate slower version # OP P = (CPREAL n, r)CPREAL: ( # alternate slower version # IF n < r ORF r < 0 THEN IF NOT cp fix value error(CPARGS("P",ENTIER n,ENTIER r)) THEN stop FI FI; CPREAL out := 1; # basically nPk = (n-r+1)(n-r+2)...(n-2)(n-1)n = n!/(n-r+1)! # CPREAL i := n-r+1; WHILE i <= n DO Line 105 ⟶ 245: END COMMENT SKIP</~~lang~~syntaxhighlight>'''File: test_combinations_and_permutations.a68'''<~~lang~~syntaxhighlight lang="algol68">#!/usr/bin/a68g --script # # -- coding: utf-8 -- # Line 154 ⟶ 294: printf(($"("g(0)" C "g(0)") = "g(0,1)$, r, first, r C first, $", "$)); printf(($"("g(0)" C "g(0)") = "g(0,1)$, r, second, r C second, $l$)) OD</~~lang~~syntaxhighlight>'''Output:''' <pre> A sample of Permutations from 1 to 12: Line 204 ⟶ 344: (1000 C 998) = 499500.0, (1000 C 969) = 76023224077705100000000000000000000000000000000000000000000.0 </pre> =={{header\|Arturo}}== <syntaxhighlight lang="arturo">printPermutations: function [s,b][ print [ "P(" ++ (to :string s) ++ ", " ++ (to :string b) ++ ") = " permutate.count.by:b @1..s ] ] printCombinations: function [s,b][ print [ "C(" ++ (to :string s) ++ ", " ++ (to :string b) ++ ") = " combine.count.by:b @1..s ] ] printPermutations 4 2 printPermutations 5 3 printPermutations 6 4 printPermutations 7 5 printPermutations 8 6 printPermutations 9 7 printPermutations 10 8 printPermutations 11 9 printPermutations 12 10 printCombinations 10 8 printCombinations 20 18 printCombinations 30 28 printCombinations 40 38 printCombinations 50 48 printCombinations 60 58 printPermutations 340 230 printPermutations 12503 150 printCombinations 180 40 printCombinations 970 730 </syntaxhighlight> {{out}} <pre>P(4, 2) = 12 P(5, 3) = 60 P(6, 4) = 360 P(7, 5) = 2520 P(8, 6) = 20160 P(9, 7) = 181440 P(10, 8) = 1814400 P(11, 9) = 19958400 P(12, 10) = 239500800 C(10, 8) = 45 C(20, 18) = 190 C(30, 28) = 435 C(40, 38) = 780 C(50, 48) = 1225 C(60, 58) = 1770 P(340, 230) = 321163468667018390278490855166116943643915115883733627394541858241858716874364895663708654024956544433526906929220370657178037367849691887585605027786348781323683175128970012088045935740649918420588744620257725216845699632806302776510600871474726786041782200035955935929612857643030896557053167285732409630111804341546261862735713347179073784452257244336775041366281305006608295780372754879670614948719662927123558774623452956076264437724405141046975318837351290035912604435611648000000000000000000000000000000000000000000000000000000000 P(12503, 150) = 145349029342353586631193854127379263889050707122840603258866143765734805000836318095430535878283222399986691383724192776832431401841845807936456562635873483263903741650446537273646795588914347285406358840689987529465482210964310466322364415189531610901593861947474126246323721536314280740814336769614384426370056684232165925069433834623283158338231715677812329656785966269970444585319639178491261742879187758508103438009278190651455732354378754778290406101258502700133885808094777181331499601845451634234055547273229226095279158651885556410705696845808314349918299242931159040000000000000000000000000000000000000000 C(180, 40) = 18290112128130495302949953479795255876590 C(970, 730) = 1460703798049835817371501066063364773159305309662093793682805325059313312798468600051903640905890411826065678898559485932184945661085268182815842989416642105002033447941384758262043195945021701492825484266010877073654424416642736902160</pre> =={{header\|Bracmat}}== This solution shows results from bignum as well as floating point implementations. If bignum answers are very big, only the first 50 digits and a count of the digits that are not shown are shown. The bignum answers are shown as Cn and Pn, while the floating point answers are shown as Cf and Pf. Answers are aligned to make it easy to compare the decimals. <syntaxhighlight lang="bracmat"> ( C = n k coef . !arg:(?n,?k) & (!n+-1!k:<!k:?k\|) & 1:?coef & whl ' ( !k:>0 & !coef!n!k^-1:?coef & !k+-1:?k & !n+-1:?n ) & !coef ) & ( compileBinomialFunctionThatDoesFloatingPointCalculations = . new $ ( UFP , ' ( (s.n) (s.k) . "************************************************************ * Notice the difference between the following four lines * * of code and the much shorter (!n+-1!k:<!k:?k\|) in * function C above. UFP grammar is simpler than usual * * Bracmat grammar. UFP code is therefore less terse. * ***********************************************************" & !n+-1!k:?n-k & ( !n-k:<!k&!n-k:?k \| ) & 1:?coef & whl ' ( !k:>0 & !coef!n!k^-1:?coef & !k+-1:?k & !n+-1:?n ) & !coef ) ) ) & compileBinomialFunctionThatDoesFloatingPointCalculations$ : ?binom & ( P = n k result . !arg:(?n,?k) & !n+-1!k:?k & 1:?result & whl ' ( !n:>!k & !n!result:?result & !n+-1:?n ) & !result ) & ( compilePermutationFunctionThatDoesFloatingPointCalculations = . new $ ( UFP , ' ( (s.n) (s.k) . !n+-1!k:?k & 1:?result & whl ' ( !n:>!k & !n!result:?result & !n+-1:?n ) & !result ) ) ) & compilePermutationFunctionThatDoesFloatingPointCalculations$ : ?permu & 0:?i & whl ' ( 1+!i:~>12:?i & div$(!i.3):?k & out$(!i P !k "=" P$(!i,!k)) ) & 0:?i & whl ' ( 10+!i:~>60:?i & div$(!i.3):?k & out$(!i Cn !k "= " C$(!i,!k)) & out$(!i Cf !k "=" (binom..go)$(!i,!k)) ) & ( displayBig = . @(!arg:?show [50 ? [?length) & !show "... (" !length+-50 " more digits)" \| !arg ) & 5 50 500 1000 5000 15000:?is & whl ' ( !is:%?i ?is & div$(!i.3):?k & out $ ( str $ (!i " Pn " !k " = " displayBig$(P$(!i,!k))) ) & out $ ( str $ (!i " Pf " !k " = " (permu..go)$(!i,!k)) ) ) & 0:?i & whl ' ( 100+!i:~>1000:?i & div$(!i.3):?k & out $ ( str $ (!i " Cn " !k " = " displayBig$(C$(!i,!k))) ) & out $ ( str $ (!i " Cf " !k " = " (binom..go)$(!i,!k)) ) ) & all done;</syntaxhighlight> Output: <pre>1 P 0 = 1 2 P 0 = 1 3 P 1 = 3 4 P 1 = 4 5 P 1 = 5 6 P 2 = 30 7 P 2 = 42 8 P 2 = 56 9 P 3 = 504 10 P 3 = 720 11 P 3 = 990 12 P 4 = 11880 10 Cn 3 = 120 10 Cf 3 = 1.1999999999999999E+02 20 Cn 6 = 38760 20 Cf 6 = 3.8759999999999993E+04 30 Cn 10 = 30045015 30 Cf 10 = 3.0045014999999989E+07 40 Cn 13 = 12033222880 40 Cf 13 = 1.2033222880000000E+10 50 Cn 16 = 4923689695575 50 Cf 16 = 4.9236896955750000E+12 60 Cn 20 = 4191844505805495 60 Cf 20 = 4.1918445058054930E+15 5 Pn 1 = 5 5 Pf 1 = 5.0000000000000000E+00 50 Pn 16 = 103017324974226408345600000 50 Pf 16 = 1.0301732497422640E+26 500 Pn 166 = 35348749217429427876093618266017623068440028791060... (385 more digits) 500 Pf 166 = INF 1000 Pn 333 = 59693262885034150890397017659007842809981894765670... (922 more digits) 1000 Pf 333 = INF 5000 Pn 1666 = 68567457572556742754845369402488960622341567102448... (5976 more digits) 5000 Pf 1666 = INF 15000 Pn 5000 = 96498539887274939220148588059312959807922816886808... (20420 more digits) 15000 Pf 5000 = INF 100 Cn 33 = 294692427022540894366527900 100 Cf 33 = 2.9469242702254079E+26 200 Cn 66 = 72697525451692783415270666511927389767550269141935... (4 more digits) 200 Cf 66 = 7.2697525451692816E+53 300 Cn 100 = 41582514632585647447833835263264055802804660057436... (32 more digits) 300 Cf 100 = 4.1582514632585569E+81 400 Cn 133 = 12579486841821087021333484756519650044917494358375... (60 more digits) 400 Cf 133 = 1.2579486841821091E+109 500 Cn 166 = 39260283861944227552204083450723314281973490135301... (87 more digits) 500 Cf 166 = 3.9260283861944195E+136 600 Cn 200 = 25060177832214028050056167705132288352025510250879... (115 more digits) 600 Cf 200 = 2.5060177832213946E+164 700 Cn 233 = 81032035633395999047404536440311382329449203119421... (142 more digits) 700 Cf 233 = 8.1032035633395859E+191 800 Cn 266 = 26456233626836270342888292995561242550915240450150... (170 more digits) 800 Cf 266 = 2.6456233626836295E+219 900 Cn 300 = 17433563732964466429607307650857183476303419689548... (198 more digits) 900 Cf 300 = 1.7433563732964451E+247 1000 Cn 333 = 57761345531476516697774863235496017223394580195002... (225 more digits) 1000 Cf 333 = 5.7761345531476355E+274 {!} all done</pre> =={{header\|C}}== Using big integers. GMP in fact has a factorial function which is quite possibly more efficient, though using it would make code longer. <syntaxhighlight lang="c">#include <gmp.h> void perm(mpz_t out, int n, int k) { mpz_set_ui(out, 1); k = n - k; while (n > k) mpz_mul_ui(out, out, n--); } void comb(mpz_t out, int n, int k) { perm(out, n, k); while (k) mpz_divexact_ui(out, out, k--); } int main(void) { mpz_t x; mpz_init(x); perm(x, 1000, 969); gmp_printf("P(1000,969) = %Zd\n", x); comb(x, 1000, 969); gmp_printf("C(1000,969) = %Zd\n", x); return 0; }</syntaxhighlight> =={{header\|C#}}== {{trans\|Java}} <syntaxhighlight lang="C#"> using System; using System.Numerics; class CombinationsAndPermutations { public static void Main(string[] args) { Console.WriteLine(double.MaxValue); Console.WriteLine("A sample of permutations from 1 to 12 with exact Integer arithmetic:"); for (int n = 1; n <= 12; n++) { int k = n / 2; Console.WriteLine($"{n} P {k} = {Permutation(n, k)}"); } Console.WriteLine(); Console.WriteLine("A sample of combinations from 10 to 60 with exact Integer arithmetic:"); for (int n = 10; n <= 60; n += 5) { int k = n / 2; Console.WriteLine($"{n} C {k} = {Combination(n, k)}"); } Console.WriteLine(); Console.WriteLine("A sample of permutations from 5 to 15000 displayed in floating point arithmetic:"); Console.WriteLine($"{5} P {2} = {Display(Permutation(5, 2), 50)}"); for (int n = 1000; n <= 15000; n += 1000) { int k = n / 2; Console.WriteLine($"{n} P {k} = {Display(Permutation(n, k), 50)}"); } Console.WriteLine(); Console.WriteLine("A sample of combinations from 100 to 1000 displayed in floating point arithmetic:"); for (int n = 100; n <= 1000; n += 100) { int k = n / 2; Console.WriteLine($"{n} C {k} = {Display(Combination(n, k), 50)}"); } } private static string Display(BigInteger val, int precision) { string s = val.ToString(); // Ensure that we don't try to take a substring longer than what's available. int actualPrecision = Math.Min(precision, s.Length - 1); // Adjusted to ensure it doesn't exceed string length System.Text.StringBuilder sb = new System.Text.StringBuilder(); if (s.Length > 1) // Check if the string has more than one character { sb.Append(s.Substring(0, 1)); sb.Append("."); sb.Append(s.Substring(1, actualPrecision-1)); } else { // If the string is only one digit, no need to insert a decimal point. sb.Append(s); } sb.Append(" * 10^"); sb.Append(s.Length - 1); return sb.ToString(); } public static BigInteger Combination(int n, int k) { // Select value with smallest intermediate results // combination(n, k) = combination(n, n-k) if (n - k < k) { k = n - k; } BigInteger result = Permutation(n, k); while (k > 0) { result = result / k; k--; } return result; } public static BigInteger Permutation(int n, int k) { BigInteger result = BigInteger.One; for (int i = n; i >= n - k + 1; i--) { result = result * i; } return result; } } </syntaxhighlight> {{out}} <pre> 1.79769313486232E+308 A sample of permutations from 1 to 12 with exact Integer arithmetic: 1 P 0 = 1 2 P 1 = 2 3 P 1 = 3 4 P 2 = 12 5 P 2 = 20 6 P 3 = 120 7 P 3 = 210 8 P 4 = 1680 9 P 4 = 3024 10 P 5 = 30240 11 P 5 = 55440 12 P 6 = 665280 A sample of combinations from 10 to 60 with exact Integer arithmetic: 10 C 5 = 252 15 C 7 = 6435 20 C 10 = 184756 25 C 12 = 5200300 30 C 15 = 155117520 35 C 17 = 4537567650 40 C 20 = 137846528820 45 C 22 = 4116715363800 50 C 25 = 126410606437752 55 C 27 = 3824345300380220 60 C 30 = 118264581564861424 A sample of permutations from 5 to 15000 displayed in floating point arithmetic: 5 P 2 = 2. * 10^1 1000 P 500 = 3.2978863640988537122024252070116261706996485697981 * 10^1433 2000 P 1000 = 8.2415012140674255266380217928953202425839550211348 * 10^3167 3000 P 1500 = 8.6229457673788561572282325050469704818001698192368 * 10^5015 4000 P 2000 = 5.5146268042637929575202252487087217092097210095831 * 10^6937 5000 P 2500 = 2.5959899179479570425022364594725223975158226787173 * 10^8914 6000 P 3000 = 6.4684674799045492726351265543133876130483115297807 * 10^10934 7000 P 3500 = 3.6978393541988590953223392044469547113117500019066 * 10^12991 8000 P 4000 = 2.8347416494109128583587900207169062803923698647730 * 10^15079 9000 P 4500 = 3.5613559022062915735922606191272180393638776777367 * 10^17194 10000 P 5000 = 6.7310091721588924046678115055013992269082753633190 * 10^19333 11000 P 5500 = 7.1714299723894354619599359891785521535675525394493 * 10^21494 12000 P 6000 = 4.4778633072110971517945389774618813158941392935141 * 10^23675 13000 P 6500 = 3.6571405647713639990172807053439865891697112165983 * 10^25874 14000 P 7000 = 1.5680448683572888099973548116924953997328821172933 * 10^28090 15000 P 7500 = 2.2454775022523692436484435950526532351441855443096 * 10^30321 A sample of combinations from 100 to 1000 displayed in floating point arithmetic: 100 C 50 = 1.0089134454556419333481249725 * 10^29 200 C 100 = 9.0548514656103281165404177077484163874504589675413 * 10^58 300 C 150 = 9.3759702772827452793193754439064084879232655700081 * 10^88 400 C 200 = 1.0295250013541443297297588032040198675721092538107 * 10^119 500 C 250 = 1.1674431578827768292093473476217661965923008118031 * 10^149 600 C 300 = 1.3510794199619426851447487797850453039723394544919 * 10^179 700 C 350 = 1.5857433585316795487607022764754299250587342835720 * 10^209 800 C 400 = 1.8804244186835312700958607615195351332156581822914 * 10^239 900 C 450 = 2.2474718820660159573188913903771345843514101350359 * 10^269 1000 C 500 = 2.7028824094543656951561469362597527549615200844654 * 10^299 </pre> =={{header\|C++}}== {{libheader\|Boost}} Compiled with <code>g++-10 -Wall -std=c++03</code>, linked with <code>-lgmp</code> <syntaxhighlight lang="cpp">#include <boost/multiprecision/gmp.hpp> #include <iostream> using namespace boost::multiprecision; mpz_int p(uint n, uint p) { mpz_int r = 1; mpz_int k = n - p; while (n > k) r = n--; return r; } mpz_int c(uint n, uint k) { mpz_int r = p(n, k); while (k) r /= k--; return r; } int main() { for (uint i = 1u; i < 12u; i++) std::cout << "P(12," << i << ") = " << p(12u, i) << std::endl; for (uint i = 10u; i < 60u; i += 10u) std::cout << "C(60," << i << ") = " << c(60u, i) << std::endl; return 0; }</syntaxhighlight> {{out}} <pre>P(12,1) = 12 P(12,2) = 132 P(12,3) = 1320 P(12,4) = 11880 P(12,5) = 95040 P(12,6) = 665280 P(12,7) = 3991680 P(12,8) = 19958400 P(12,9) = 79833600 P(12,10) = 239500800 P(12,11) = 479001600 C(60,10) = 75394027566 C(60,20) = 4191844505805495 C(60,30) = 118264581564861424 C(60,40) = 4191844505805495 C(60,50) = 75394027566</pre> =={{header\|Common Lisp}}== <syntaxhighlight lang="lisp">(defun combinations (n k) (cond ((or (< n k) (< k 0) (< n 0)) 0) ((= k 0) 1) (t (do ((i 1 (1+ i)) (m n (1- m)) (a m (* a m)) (b i (* b i))) ((= i k) (/ a b)))))) (defun permutations (n k) (cond ((or (< n k) (< k 0) (< n 0)) 0) ((= k 0) 1) (t (do* ((i 1 (1+ i)) (m n (1- m)) (a m (* a m))) ((= i k) a))))) </syntaxhighlight> =={{header\|Crystal}}== {{trans\|Ruby}} <syntaxhighlight lang="ruby">require "big" include Math struct Int def permutation(k) (self-k+1..self).product(1.to_big_i) end def combination(k) self.permutation(k) // (1..k).product(1.to_big_i) end def big_permutation(k) exp(lgamma_plus(self) - lgamma_plus(self-k)) end def big_combination(k) exp( lgamma_plus(self) - lgamma_plus(self - k) - lgamma_plus(k)) end private def lgamma_plus(n) lgamma(n+1) #lgamma is the natural log of gamma end end p 12.permutation(9) #=> 79833600 p 12.big_permutation(9) #=> 79833600.00000021 p 60.combination(53) #=> 386206920 p 145.big_permutation(133) #=> 1.6801459655817956e+243 p 900.big_combination(450) #=> 2.247471882064647e+269 p 1000.big_combination(969) #=> 7.602322407770517e+58 p 15000.big_permutation(73) #=> 6.004137561717704e+304 #That's about the maximum of Float: p 15000.big_permutation(74) #=> Infinity #Fixnum has no maximum: p 15000.permutation(74) #=> 896237613852967826239917238565433149353074416025197784301593335243699358040738127950872384197159884905490054194835376498534786047382445592358843238688903318467070575184552953997615178973027752714539513893159815472948987921587671399790410958903188816684444202526779550201576117111844818124800000000000000000000 </syntaxhighlight> {{out}} <pre> 79833600 79833600.00000021 386206920 1.6801459655817956e+243 2.247471882064647e+269 7.602322407770517e+58 6.004137561717704e+304 Infinity 896237613852967826239917238565433149353074416025197784301593335243699358040738127950872384197159884905490054194835376498534786047382445592358843238688903318467070575184552953997615178973027752714539513893159815472948987921587671399790410958903188816684444202526779550201576117111844818124800000000000000000000 </pre> =={{header\|D}}== {{trans\|Ruby}} <syntaxhighlight lang="d">import std.stdio, std.mathspecial, std.range, std.algorithm, std.bigint, std.conv; enum permutation = (in uint n, in uint k) pure => reduce!q{a * b}(1.BigInt, iota(n - k + 1, n + 1)); enum combination = (in uint n, in uint k) pure => n.permutation(k) / reduce!q{a * b}(1.BigInt, iota(1, k + 1)); enum bigPermutation = (in uint n, in uint k) => exp(logGamma(n + 1) - logGamma(n - k + 1)); enum bigCombination = (in uint n, in uint k) => exp(logGamma(n + 1) - logGamma(n - k + 1) - logGamma(k + 1)); void main() { 12.permutation(9).writeln; 12.bigPermutation(9).writeln; 60.combination(53).writeln; 145.bigPermutation(133).writeln; 900.bigCombination(450).writeln; 1_000.bigCombination(969).writeln; 15_000.bigPermutation(73).writeln; 15_000.bigPermutation(1185).writeln; writefln("%(%s\\\n%)", 15_000.permutation(74).text.chunks(50)); }</syntaxhighlight> {{out}} <pre>79833600 7.98336e+07 386206920 1.68015e+243 2.24747e+269 7.60232e+58 6.00414e+304 6.31335e+4927 89623761385296782623991723856543314935307441602519\ 77843015933352436993580407381279508723841971598849\ 05490054194835376498534786047382445592358843238688\ 90331846707057518455295399761517897302775271453951\ 38931598154729489879215876713997904109589031888166\ 84444202526779550201576117111844818124800000000000\ 000000000</pre> =={{header\|EasyLang}}== {{trans\|Phix}} <syntaxhighlight> func perm x y . z = 1 for i = x - y + 1 to x z = i . return z . func fact x . z = 1 for i = 2 to x z = i . return z . func comb x y . if x - y < y y = x - y . return perm x y / fact y . # e = 2.7182818284590452354 func log n . return log10 n / log10 e . func lstirling n . if n < 10 return lstirling (n + 1) - log (n + 1) . return 0.5 * log (2 * pi * n) + n * log (n / e + 1 / (12 * e * n)) . func$ tolog v . h = v div log 10 return pow e (v - h * log 10) & "e" & h . func$ permf n k . return tolog (lstirling n - lstirling (n - k)) . func$ combf n k . return tolog (lstirling n - lstirling (n - k) - lstirling k) . print "=> Exact results:" for n = 1 to 12 p = n div 3 print "P(" & n & "," & p & ")=" & perm n p . # # double has 53 bits for integer # for n = 10 step 10 to 50 p = n div 3 print "C(" & n & "," & p & ")=" & comb n p . # print "" print "=> Floating point approximations:" for n in [ 5 50 500 1000 5000 15000 ] p = n div 3 print "P(" & n & "," & p & ")=" & permf n p . for n = 100 step 100 to 1000 p = n div 3 print "C(" & n & "," & p & ")=" & combf n p . </syntaxhighlight> =={{header\|EchoLisp}}== <syntaxhighlight lang="scheme"> ;; rename native functions according to task (define-syntax-rule (Cnk n k) (Cnp n k)) (define-syntax-rule (Ank n k) (Anp n k)) (Cnk 10 1) → 10 (lib 'bigint) ;; no floating point needed : use large integers (Cnk 100 10) → 17310309456440 (Cnk 1000 42) → 297242911333923795640059429176065863139989673213703918037987737481286092000 (Ank 10 10) → 3628800 (factorial 10) → 3628800 (Ank 666 42) → 1029024198692120734765388598788124551227594950478035495578451793852872815678512303375588360 1398831219998720000000000000 </syntaxhighlight> =={{header\|Elixir}}== {{trans\|Erlang}} <syntaxhighlight lang="elixir">defmodule Combinations_permutations do def perm(n, k), do: product(n - k + 1 .. n) def comb(n, k), do: div( perm(n, k), product(1 .. k) ) defp product(a..b) when a>b, do: 1 defp product(list), do: Enum.reduce(list, 1, fn n, acc -> n * acc end) def test do IO.puts "\nA sample of permutations from 1 to 12:" Enum.each(1..12, &show_perm(&1, div(&1, 3))) IO.puts "\nA sample of combinations from 10 to 60:" Enum.take_every(10..60, 10) \|> Enum.each(&show_comb(&1, div(&1, 3))) IO.puts "\nA sample of permutations from 5 to 15000:" Enum.each([5,50,500,1000,5000,15000], &show_perm(&1, div(&1, 3))) IO.puts "\nA sample of combinations from 100 to 1000:" Enum.take_every(100..1000, 100) \|> Enum.each(&show_comb(&1, div(&1, 3))) end defp show_perm(n, k), do: show_gen(n, k, "perm", &perm/2) defp show_comb(n, k), do: show_gen(n, k, "comb", &comb/2) defp show_gen(n, k, strfun, fun), do: IO.puts "#{strfun}(#{n}, #{k}) = #{show_big(fun.(n, k), 40)}" defp show_big(n, limit) do strn = to_string(n) if String.length(strn) < limit do strn else {shown, hidden} = String.split_at(strn, limit) "#{shown}... (#{String.length(hidden)} more digits)" end end end Combinations_permutations.test</syntaxhighlight> {{out}} <pre> A sample of permutations from 1 to 12: perm(1, 0) = 1 perm(2, 0) = 1 perm(3, 1) = 3 perm(4, 1) = 4 perm(5, 1) = 5 perm(6, 2) = 30 perm(7, 2) = 42 perm(8, 2) = 56 perm(9, 3) = 504 perm(10, 3) = 720 perm(11, 3) = 990 perm(12, 4) = 11880 A sample of combinations from 10 to 60: comb(10, 3) = 120 comb(20, 6) = 38760 comb(30, 10) = 30045015 comb(40, 13) = 12033222880 comb(50, 16) = 4923689695575 comb(60, 20) = 4191844505805495 A sample of permutations from 5 to 15000: perm(5, 1) = 5 perm(50, 16) = 103017324974226408345600000 perm(500, 166) = 3534874921742942787609361826601762306844... (395 more digits) perm(1000, 333) = 5969326288503415089039701765900784280998... (932 more digits) perm(5000, 1666) = 6856745757255674275484536940248896062234... (5986 more digits) perm(15000, 5000) = 9649853988727493922014858805931295980792... (20430 more digits) A sample of combinations from 100 to 1000: comb(100, 33) = 294692427022540894366527900 comb(200, 66) = 7269752545169278341527066651192738976755... (14 more digits) comb(300, 100) = 4158251463258564744783383526326405580280... (42 more digits) comb(400, 133) = 1257948684182108702133348475651965004491... (70 more digits) comb(500, 166) = 3926028386194422755220408345072331428197... (97 more digits) comb(600, 200) = 2506017783221402805005616770513228835202... (125 more digits) comb(700, 233) = 8103203563339599904740453644031138232944... (152 more digits) comb(800, 266) = 2645623362683627034288829299556124255091... (180 more digits) comb(900, 300) = 1743356373296446642960730765085718347630... (208 more digits) comb(1000, 333) = 5776134553147651669777486323549601722339... (235 more digits) </pre> =={{header\|Erlang}}== {{trans\|Haskell}} <syntaxhighlight lang="erlang"> -module(combinations_permutations). -export([test/0]). perm(N, K) -> product(lists:seq(N - K + 1, N)). comb(N, K) -> perm(N, K) div product(lists:seq(1, K)). product(List) -> lists:foldl(fun(N, Acc) -> N * Acc end, 1, List). test() -> io:format("\nA sample of permutations from 1 to 12:\n"), [show_perm({N, N div 3}) \|\| N <- lists:seq(1, 12)], io:format("\nA sample of combinations from 10 to 60:\n"), [show_comb({N, N div 3}) \|\| N <- lists:seq(10, 60, 10)], io:format("\nA sample of permutations from 5 to 15000:\n"), [show_perm({N, N div 3}) \|\| N <- [5,50,500,1000,5000,15000]], io:format("\nA sample of combinations from 100 to 1000:\n"), [show_comb({N, N div 3}) \|\| N <- lists:seq(100, 1000, 100)], ok. show_perm({N, K}) -> show_gen(N, K, "perm", fun perm/2). show_comb({N, K}) -> show_gen(N, K, "comb", fun comb/2). show_gen(N, K, StrFun, Fun) -> io:format("~s(~p, ~p) = ~s\n",[StrFun, N, K, show_big(Fun(N, K), 40)]). show_big(N, Limit) -> StrN = integer_to_list(N), case length(StrN) < Limit of true -> StrN; false -> {Shown, Hidden} = lists:split(Limit, StrN), io_lib:format("~s... (~p more digits)", [Shown, length(Hidden)]) end. </syntaxhighlight> Output: <pre style="font-size:80%"> A sample of permutations from 1 to 12: perm(1, 0) = 1 perm(2, 0) = 1 perm(3, 1) = 3 perm(4, 1) = 4 perm(5, 1) = 5 perm(6, 2) = 30 perm(7, 2) = 42 perm(8, 2) = 56 perm(9, 3) = 504 perm(10, 3) = 720 perm(11, 3) = 990 perm(12, 4) = 11880 A sample of combinations from 10 to 60: comb(10, 3) = 120 comb(20, 6) = 38760 comb(30, 10) = 30045015 comb(40, 13) = 12033222880 comb(50, 16) = 4923689695575 comb(60, 20) = 4191844505805495 A sample of permutations from 5 to 15000: perm(5, 1) = 5 perm(50, 16) = 103017324974226408345600000 perm(500, 166) = 3534874921742942787609361826601762306844... (395 more digits) perm(1000, 333) = 5969326288503415089039701765900784280998... (932 more digits) perm(5000, 1666) = 6856745757255674275484536940248896062234... (5986 more digits) perm(15000, 5000) = 9649853988727493922014858805931295980792... (20430 more digits) A sample of combinations from 100 to 1000: comb(100, 33) = 294692427022540894366527900 comb(200, 66) = 7269752545169278341527066651192738976755... (14 more digits) comb(300, 100) = 4158251463258564744783383526326405580280... (42 more digits) comb(400, 133) = 1257948684182108702133348475651965004491... (70 more digits) comb(500, 166) = 3926028386194422755220408345072331428197... (97 more digits) comb(600, 200) = 2506017783221402805005616770513228835202... (125 more digits) comb(700, 233) = 8103203563339599904740453644031138232944... (152 more digits) comb(800, 266) = 2645623362683627034288829299556124255091... (180 more digits) comb(900, 300) = 1743356373296446642960730765085718347630... (208 more digits) comb(1000, 333) = 5776134553147651669777486323549601722339... (235 more digits) </pre> =={{header\|Factor}}== As with Racket, these operations are built in and Factor has unlimited integers. <syntaxhighlight lang="factor">USING: math.combinatorics prettyprint ; 1000 10 nCk . ! 263409560461970212832400 1000 10 nPk . ! 955860613004397508326213120000</syntaxhighlight> =={{header\|FreeBASIC}}== {{trans\|Visual Basic .NET}} <syntaxhighlight lang="freebasic">Function PermBig(x As Long, y As Long) As ULongint Dim As Long i Dim As Longint z = 1 For i = x - y + 1 To x z = z * i Next i Return (z) End Function Function FactBig(x As Long) As ULongint Dim As Long i Dim As Longint z = 1 For i = 2 To x z = z * i Next i Return (z) End Function Function CombBig(Byval x As Long, Byval y As Long) As Double If y > x Then Return (0) Elseif x = y Then Return (1) Else If x - y < y Then y = x - y Return (PermBig(x, y) / FactBig(y)) End If End Function Dim As Long i, j Print "-- Long Integer - Permutations - from 1 to 12" For i = 1 To 12 For j = 1 To i Print "P(" & i & "," & j & ")=" & Str(PermBig(i, j)) & " "; Next j Print "" Next i Print Chr(10) & "-- Float integer - Combinations from 10 to 60" For i = 10 To 60 Step 10 For j = 1 To i Step i \ 5 Print "C(" & i & "," & j & ")=" & Str(CombBig(i, j)) & " "; Next j Print "" Next i Print Chr(10) & "-- Float integer - Permutations from 5000 to 15000" For i = 5000 To 15000 Step 5000 For j = 10 To 50 Step 20 Print "P(" & i & "," & j & ")=" & Str(PermBig(i, j)) & " "; Next j Print "" Next i Print Chr(10) & "-- Float integer - Combinations from 200 to 1000" For i = 200 To 1000 Step 200 For j = 20 To 100 Step 20 Print "C(" & i & "," & j & ")=" & Str(CombBig(i, j)) & " "; Next j Print "" Next i Sleep</syntaxhighlight> =={{header\|Go}}== Go has arbitrary-length maths in the standard math/big library; no need for floating-point approximations at this level. <syntaxhighlight lang="go"> package main import ( "fmt" "math/big" ) func main() { var n, p int64 fmt.Printf("A sample of permutations from 1 to 12:\n") for n = 1; n < 13; n++ { p = n / 3 fmt.Printf("P(%d,%d) = %d\n", n, p, perm(big.NewInt(n), big.NewInt(p))) } fmt.Printf("\nA sample of combinations from 10 to 60:\n") for n = 10; n < 61; n += 10 { p = n / 3 fmt.Printf("C(%d,%d) = %d\n", n, p, comb(big.NewInt(n), big.NewInt(p))) } fmt.Printf("\nA sample of permutations from 5 to 15000:\n") nArr := [...]int64{5, 50, 500, 1000, 5000, 15000} for _, n = range nArr { p = n / 3 fmt.Printf("P(%d,%d) = %d\n", n, p, perm(big.NewInt(n), big.NewInt(p))) } fmt.Printf("\nA sample of combinations from 100 to 1000:\n") for n = 100; n < 1001; n += 100 { p = n / 3 fmt.Printf("C(%d,%d) = %d\n", n, p, comb(big.NewInt(n), big.NewInt(p))) } } func fact(n big.Int) big.Int { if n.Sign() < 1 { return big.NewInt(0) } r := big.NewInt(1) i := big.NewInt(2) for i.Cmp(n) < 1 { r.Mul(r, i) i.Add(i, big.NewInt(1)) } return r } func perm(n, k big.Int) big.Int { r := fact(n) r.Div(r, fact(n.Sub(n, k))) return r } func comb(n, r big.Int) big.Int { if r.Cmp(n) == 1 { return big.NewInt(0) } if r.Cmp(n) == 0 { return big.NewInt(1) } c := fact(n) den := fact(n.Sub(n, r)) den.Mul(den, fact(r)) c.Div(c, den) return c } </syntaxhighlight> Output: <pre>A sample of permutations from 1 to 12: P(1,0) = 1 P(2,0) = 1 P(3,1) = 3 P(4,1) = 4 P(5,1) = 5 P(6,2) = 30 P(7,2) = 42 P(8,2) = 56 P(9,3) = 504 P(10,3) = 720 P(11,3) = 990 P(12,4) = 11880 A sample of combinations from 10 to 60: C(10,3) = 120 C(20,6) = 38760 C(30,10) = 30045015 C(40,13) = 12033222880 C(50,16) = 4923689695575 C(60,20) = 4191844505805495 A sample of permutations from 5 to 15000: P(5,1) = 5 P(50,16) = 103017324974226408345600000 P(500,166) = 353487492174294278760936182660176230684400287910609032932176169251145051223056590013516735448538086130105926216996155913554025250125337170813019594283712354977999534430770809915541863294344717641926832713607917635838943102385935821177067602075180371673503765613359169210620516084434587852075431010684540863423686685437846488590916158047347611875355166780660833377163468853354607169353747005440000000000000000000000000000000000000000000 P(1000,333) = 596932628850341508903970176590078428099818947656708993181003187015748137551596090897395098770177006143741943000028735297176540996715216223470117008188290845824893667956971794591724165149886070514986743432208287422668258597883938335639789463537748828480742878588232053442156529356123644254034620250998013239063050800571901268462622323786888857845397534124006543754044331948142416200311556601875197681493636464229808874876131434533721937546154413110195799580966669097512255452752067457629468146295647406985227990184437533735467426422585063839734564755495294687791091277426861971582062496498138090957027659529000216472781766770467939751274131497380920782541878343751496719223481266229644276166815775979972070711062842180838624567600323874889368489920534418201154996144776696986749038638993937273035934558675329887334851616951614782042039326589303359315137179445599073086196250614391361241088000000000000000000000000000000000000000000000000000000000000000000000000000000000000 P(5000,1666) = 685674575725567427548453694024889606223415671024481949274389525763165050940278456955930987049278913288741555958551429473866794447094151777406284837935079289013939976368320856320075398810321021264609367677596630017492662476653457696667818345302032079158497998678962485919859205035598026552321633034034950067207086196635562039504231368320682342175536748176816241129369211418155064764285518551122913802314264757797230065345837585112169021234001053819461909997341171332285483908469793127022101882927191957584158820572953454348525922376264896938701700503579[cut for brevity] P(15000,5000) = 964985398872749392201485880593129598079228168868086089709069882934904327326041434448685068557423519588782990996839148935743422697293448340149394869739166392123184483900811562453898657822593544251663672426442538772790754374376708208571908721887559582227891355612508675254239851818491849618848602192999811776790423150460291684931309968390568370095954400889615967984376006072547506893620140168365357740946620575769516153753735678390720704316317251046076250691800213698137901479234159657387339301806781869228964210268691780310799227446113809960921205463002689680686581917548426454723087176124161728203856922785131850458595700123281745279678944233081992533251647291980003523846698046974475328766195265764029672828687155439288711515137638802309221117722696141292043413736618293499869841412502567093112620576877254953509763190120514494690844850900732795773648193199871673637389424514221019995024902360799221264651063770768194819355718580618775786102220736660007307021195929713637615667876228769039112504881578385857681786265840651390718446642041061982261630233585694136722143048873223923363402206453540375800805584706454386378553117095912890292078608807203370350094921116703330118357576158786202069866488676813049328910886961677198798213135372964228023956951408480830806905842004749290253667472325116488994202675719755525469268037451046892994419861436389890344224919839492952550142350743[cut for brevity] A sample of combinations from 100 to 1000: C(100,33) = 294692427022540894366527900 C(200,66) = 726975254516927834152706665119273897675502691419359300 C(300,100) = 4158251463258564744783383526326405580280466005743648708663033657304756328324008620 C(400,133) = 12579486841821087021333484756519650044917494358375678689903944407062661887691782714561293494166382210572895600 C(500,166) = 39260283861944227552204083450723314281973490135301528555644859907093815055309467400410307687655531369748267877388321068535356654999949000 C(600,200) = 250601778322140280500561677051322883520255102508793389473094343332441398315528846878026090182866148274621477126087990864486283260622128340138769443055475567389095596 C(700,233) = 810320356333959990474045364403113823294492031194214481466666874664632951279413378341573227990559808332117096088495529108312831240206642074673862825014526456696556162909686658807978793453220000 C(800,266) = 2645623362683627034288829299556124255091524045015061559880110850588357798837813526621954238671844949835878984342140544523564918954064307529607727040078866833961879433579846596361407958192581870170248962672479120257018000 C(900,300) = 17433563732964466429607307650857183476303419689548896834573896609295907147982188408607179689430757813632301567003238290404591375673438892792913992016448098783043956457942357838233534288303678577768883350818012950587783262800434058273110416350965200 C(1000,333) = 57761345531476516697774863235496017223394580195002114171799054793660405129610824218694609475292335509897216233568933163108481350037180876217607177657236327948642711456536116349650593787069554795812874358426845137087373717847050642650744775784499136594696491030795647099932000 </pre> =={{header\|Haskell}}== The Haskell Integer type supports arbitrary precision so floating point approximation is not needed. <syntaxhighlight lang="haskell">perm :: Integer -> Integer -> Integer perm n k = product [n-k+1..n] comb :: Integer -> Integer -> Integer comb n k = perm n k `div` product [1..k] main :: IO () main = do let showBig maxlen b = let st = show b stlen = length st in if stlen < maxlen then st else take maxlen st ++ "... (" ++ show (stlen-maxlen) ++ " more digits)" let showPerm pr = putStrLn $ "perm(" ++ show n ++ "," ++ show k ++ ") = " ++ showBig 40 (perm n k) where n = fst pr k = snd pr let showComb pr = putStrLn $ "comb(" ++ show n ++ "," ++ show k ++ ") = " ++ showBig 40 (comb n k) where n = fst pr k = snd pr putStrLn "A sample of permutations from 1 to 12:" mapM_ showPerm [(n, n `div` 3) \| n <- [1..12] ] putStrLn "" putStrLn "A sample of combinations from 10 to 60:" mapM_ showComb [(n, n `div` 3) \| n <- [10,20..60] ] putStrLn "" putStrLn "A sample of permutations from 5 to 15000:" mapM_ showPerm [(n, n `div` 3) \| n <- [5,50,500,1000,5000,15000] ] putStrLn "" putStrLn "A sample of combinations from 100 to 1000:" mapM_ showComb [(n, n `div` 3) \| n <- [100,200..1000] ] </syntaxhighlight> {{out}} <pre style="font-size:80%">A sample of permutations from 1 to 12: perm(1,0) = 1 perm(2,0) = 1 perm(3,1) = 3 perm(4,1) = 4 perm(5,1) = 5 perm(6,2) = 30 perm(7,2) = 42 perm(8,2) = 56 perm(9,3) = 504 perm(10,3) = 720 perm(11,3) = 990 perm(12,4) = 11880 A sample of combinations from 10 to 60: comb(10,3) = 120 comb(20,6) = 38760 comb(30,10) = 30045015 comb(40,13) = 12033222880 comb(50,16) = 4923689695575 comb(60,20) = 4191844505805495 A sample of permutations from 5 to 15000: perm(5,1) = 5 perm(50,16) = 103017324974226408345600000 perm(500,166) = 3534874921742942787609361826601762306844... (395 more digits) perm(1000,333) = 5969326288503415089039701765900784280998... (932 more digits) perm(5000,1666) = 6856745757255674275484536940248896062234... (5986 more digits) perm(15000,5000) = 9649853988727493922014858805931295980792... (20430 more digits) A sample of combinations from 100 to 1000: comb(100,33) = 294692427022540894366527900 comb(200,66) = 7269752545169278341527066651192738976755... (14 more digits) comb(300,100) = 4158251463258564744783383526326405580280... (42 more digits) comb(400,133) = 1257948684182108702133348475651965004491... (70 more digits) comb(500,166) = 3926028386194422755220408345072331428197... (97 more digits) comb(600,200) = 2506017783221402805005616770513228835202... (125 more digits) comb(700,233) = 8103203563339599904740453644031138232944... (152 more digits) comb(800,266) = 2645623362683627034288829299556124255091... (180 more digits) comb(900,300) = 1743356373296446642960730765085718347630... (208 more digits) comb(1000,333) = 5776134553147651669777486323549601722339... (235 more digits) </pre> =={{header\|Icon}} and {{header\|Unicon}}== As with several other languages here, Icon and Unicon can handle unlimited integers so floating point approximation isn't needed. The sample here gives a few representative values to shorten the output. <syntaxhighlight lang="unicon">procedure main() write("P(4,2) = ",P(4,2)) write("P(8,2) = ",P(8,2)) write("P(10,8) = ",P(10,8)) write("C(10,8) = ",C(10,8)) write("C(20,8) = ",C(20,8)) write("C(60,58) = ",C(60,58)) write("P(1000,10) = ",P(1000,10)) write("P(1000,20) = ",P(1000,20)) write("P(15000,2) = ",P(15000,2)) write("C(1000,10) = ",C(1000,10)) write("C(1000,999) = ",C(1000,999)) write("C(1000,1000) = ",C(1000,1000)) write("C(15000,14998) = ",C(15000,14998)) end procedure C(n,k) every (d:=1) := 2 to k return P(n,k)/d end procedure P(n,k) every (p:=1) := (n-k+1) to n return p end</syntaxhighlight> Output: <pre> ->cap P(4,2) = 12 P(8,2) = 56 P(10,8) = 1814400 C(10,8) = 45 C(20,8) = 125970 C(60,58) = 1770 P(1000,10) = 955860613004397508326213120000 P(1000,20) = 825928413359200443640727373872992573951185652339949568000000 P(15000,2) = 224985000 C(1000,10) = 263409560461970212832400 C(1000,999) = 1000 C(1000,1000) = 1 C(15000,14998) = 112492500 -> </pre> =={{header\|J}}== It looks like this task wants a count of the available combinations or permutations (given a set of 3 things, there are three distinct combinations of 2 of them) rather than a representation of the available combinations or permutations (given a set of three things, the distinct combinations of 2 of them may be identified by <0,1>, <0,2> and <1,2>)). Also, this task allows a language to show off its abilities to support floating point numbers outside the usual range of 64 bit IEEE floating point numbers. We'll neglect that part. Implementation: <syntaxhighlight lang="j">C=: ! P=: (%&!&x:~ * <:)"0</syntaxhighlight> ! is a primitive, but we will give it a name (<code>C</code>) for this task. Example use (P is permutations, C is combinations): <syntaxhighlight lang="j"> P table 1+i.12 ┌──┬─────────────────────────────────────────────────────────────┐ │P │1 2 3 4 5 6 7 8 9 10 11 12│ ├──┼─────────────────────────────────────────────────────────────┤ │ 1│1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600│ │ 2│0 1 3 12 60 360 2520 20160 181440 1814400 19958400 239500800│ │ 3│0 0 1 4 20 120 840 6720 60480 604800 6652800 79833600│ │ 4│0 0 0 1 5 30 210 1680 15120 151200 1663200 19958400│ │ 5│0 0 0 0 1 6 42 336 3024 30240 332640 3991680│ │ 6│0 0 0 0 0 1 7 56 504 5040 55440 665280│ │ 7│0 0 0 0 0 0 1 8 72 720 7920 95040│ │ 8│0 0 0 0 0 0 0 1 9 90 990 11880│ │ 9│0 0 0 0 0 0 0 0 1 10 110 1320│ │10│0 0 0 0 0 0 0 0 0 1 11 132│ │11│0 0 0 0 0 0 0 0 0 0 1 12│ │12│0 0 0 0 0 0 0 0 0 0 0 1│ └──┴─────────────────────────────────────────────────────────────┘ C table 10+10i.6x ┌──┬─────────────────────────────────────────────────────────────────┐ │C │10 20 30 40 50 60│ ├──┼─────────────────────────────────────────────────────────────────┤ │10│ 1 184756 30045015 847660528 10272278170 75394027566│ │20│ 0 1 30045015 137846528820 47129212243960 4191844505805495│ │30│ 0 0 1 847660528 47129212243960 118264581564861424│ │40│ 0 0 0 1 10272278170 4191844505805495│ │50│ 0 0 0 0 1 75394027566│ │60│ 0 0 0 0 0 1│ └──┴─────────────────────────────────────────────────────────────────┘ 5 P 100 7.77718e155 100 P 200 8.45055e216 300 P 400 2.09224e254 700 P 800 3.18349e287 5 C 100 75287520 100 C 200 9.05485e58 300 C 400 2.24185e96 700 C 800 3.41114e129</syntaxhighlight> =={{header\|Java}}== The second part of this task implies that the computations are performed in floating point arithmetic. However, the maximum value of a double in Java is 1.7976931348623157E308. Hence, larger computations would overflow. As a result, this task was interpreted so that “using” means “displayed”. <syntaxhighlight lang="java"> import java.math.BigInteger; public class CombinationsAndPermutations { public static void main(String[] args) { System.out.println(Double.MAX_VALUE); System.out.println("A sample of permutations from 1 to 12 with exact Integer arithmetic:"); for ( int n = 1 ; n <= 12 ; n++ ) { int k = n / 2; System.out.printf("%d P %d = %s%n", n, k, permutation(n, k)); } System.out.println(); System.out.println("A sample of combinations from 10 to 60 with exact Integer arithmetic:"); for ( int n = 10 ; n <= 60 ; n += 5 ) { int k = n / 2; System.out.printf("%d C %d = %s%n", n, k, combination(n, k)); } System.out.println(); System.out.println("A sample of permutations from 5 to 15000 displayed in floating point arithmetic:"); System.out.printf("%d P %d = %s%n", 5, 2, display(permutation(5, 2), 50)); for ( int n = 1000 ; n <= 15000 ; n += 1000 ) { int k = n / 2; System.out.printf("%d P %d = %s%n", n, k, display(permutation(n, k), 50)); } System.out.println(); System.out.println("A sample of combinations from 100 to 1000 displayed in floating point arithmetic:"); for ( int n = 100 ; n <= 1000 ; n += 100 ) { int k = n / 2; System.out.printf("%d C %d = %s%n", n, k, display(combination(n, k), 50)); } } private static String display(BigInteger val, int precision) { String s = val.toString(); precision = Math.min(precision, s.length()); StringBuilder sb = new StringBuilder(); sb.append(s.substring(0, 1)); sb.append("."); sb.append(s.substring(1, precision)); sb.append(" 10^"); sb.append(s.length()-1); return sb.toString(); } public static BigInteger combination(int n, int k) { // Select value with smallest intermediate results // combination(n, k) = combination(n, n-k) if ( n-k < k ) { k = n-k; } BigInteger result = permutation(n, k); while ( k > 0 ) { result = result.divide(BigInteger.valueOf(k)); k--; } return result; } public static BigInteger permutation(int n, int k) { BigInteger result = BigInteger.ONE; for ( int i = n ; i >= n-k+1 ; i-- ) { result = result.multiply(BigInteger.valueOf(i)); } return result; } } </syntaxhighlight> {{out}} <pre> 1.7976931348623157E308 A sample of permutations from 1 to 12 with exact Integer arithmetic: 1 P 0 = 1 2 P 1 = 2 3 P 1 = 3 4 P 2 = 12 5 P 2 = 20 6 P 3 = 120 7 P 3 = 210 8 P 4 = 1680 9 P 4 = 3024 10 P 5 = 30240 11 P 5 = 55440 12 P 6 = 665280 A sample of combinations from 10 to 60 with exact Integer arithmetic: 10 C 5 = 252 15 C 7 = 6435 20 C 10 = 184756 25 C 12 = 5200300 30 C 15 = 155117520 35 C 17 = 4537567650 40 C 20 = 137846528820 45 C 22 = 4116715363800 50 C 25 = 126410606437752 55 C 27 = 3824345300380220 60 C 30 = 118264581564861424 A sample of permutations from 5 to 15000 displayed in floating point arithmetic: 5 P 2 = 2.0 * 10^1 1000 P 500 = 3.2978863640988537122024252070116261706996485697981 * 10^1433 2000 P 1000 = 8.2415012140674255266380217928953202425839550211348 * 10^3167 3000 P 1500 = 8.6229457673788561572282325050469704818001698192368 * 10^5015 4000 P 2000 = 5.5146268042637929575202252487087217092097210095831 * 10^6937 5000 P 2500 = 2.5959899179479570425022364594725223975158226787173 * 10^8914 6000 P 3000 = 6.4684674799045492726351265543133876130483115297807 * 10^10934 7000 P 3500 = 3.6978393541988590953223392044469547113117500019066 * 10^12991 8000 P 4000 = 2.8347416494109128583587900207169062803923698647730 * 10^15079 9000 P 4500 = 3.5613559022062915735922606191272180393638776777367 * 10^17194 10000 P 5000 = 6.7310091721588924046678115055013992269082753633190 * 10^19333 11000 P 5500 = 7.1714299723894354619599359891785521535675525394493 * 10^21494 12000 P 6000 = 4.4778633072110971517945389774618813158941392935141 * 10^23675 13000 P 6500 = 3.6571405647713639990172807053439865891697112165983 * 10^25874 14000 P 7000 = 1.5680448683572888099973548116924953997328821172933 * 10^28090 15000 P 7500 = 2.2454775022523692436484435950526532351441855443096 * 10^30321 A sample of combinations from 100 to 1000 displayed in floating point arithmetic: 100 C 50 = 1.00891344545564193334812497256 * 10^29 200 C 100 = 9.0548514656103281165404177077484163874504589675413 * 10^58 300 C 150 = 9.3759702772827452793193754439064084879232655700081 * 10^88 400 C 200 = 1.0295250013541443297297588032040198675721092538107 * 10^119 500 C 250 = 1.1674431578827768292093473476217661965923008118031 * 10^149 600 C 300 = 1.3510794199619426851447487797850453039723394544919 * 10^179 700 C 350 = 1.5857433585316795487607022764754299250587342835720 * 10^209 800 C 400 = 1.8804244186835312700958607615195351332156581822914 * 10^239 900 C 450 = 2.2474718820660159573188913903771345843514101350359 * 10^269 1000 C 500 = 2.7028824094543656951561469362597527549615200844654 * 10^299 </pre> =={{header\|jq}}== The C implementation of jq approximates large integers by IEEE 754 64-bit floats, and only supports tgamma (true gamma), and thus beyond about 1e308 all accuracy is lost. On the other hand, gojq, the Go implementation, supports unbounded-precision integer arithmetic, and the `permutation` and `combination` functions below are implemented so as to preserve precision. <syntaxhighlight lang="jq"> def permutation(k): . as $n \| reduce range($n-k+1; 1+$n) as $i (1; . * $i); def combination(k): . as $n \| if k > ($n/2) then combination($n-k) else reduce range(0; k) as $i (1; (. * ($n - $i)) / ($i + 1)) end; # natural log of n! def log_factorial: (1+.) \| tgamma \| log; def log_permutation(k): (log_factorial - ((.-k) \| log_factorial)); def log_combination(k): (log_factorial - ((. - k)\|log_factorial) - (k\|log_factorial)); def big_permutation(k): log_permutation(k) \| exp; def big_combination(k): log_combination(k) \| exp; </syntaxhighlight> '''Examples using the C implementation''': <pre> 12 \| permutation(9) #=> 79833600 12 \| big_permutation(9) #=> 79833599.99999964 60 \| combination(53) #=> 386206920 60 \| big_combination(53) #=> 386206920.0000046 145 \| big_permutation(133) #=> 1.6801459655817956e+243 170 \| big_permutation(133) #=> 5.272846415658284e+263 </pre> '''Examples using gojq, the Go implementation''' <pre> 60 \| combinations(30) #=> 118264581564861424 1000 \| combination(333) \| tostring \| "\(.[:40]) ... (\(length) digits in all)" #=> 5776134553147651669777486323549601722339 ... (275 digits in all) </pre> =={{header\|Julia}}== Infix operators are interpreted by Julia's parser, and (to my knowledge) it isn't possible to define arbitrary characters as such operators. However one can define Unicode "Symbol, Math" characters as infix operators. This solution uses ⊞ for combinations and ⊠ for permutations. An alternative to using the <tt>FloatingPoint</tt> versions of these operators for large numbers would be to use arbitrary precision integers, <tt>BigInt</tt>. '''Functions''' <syntaxhighlight lang="julia"> function Base.binomial{T<:FloatingPoint}(n::T, k::T) exp(lfact(n) - lfact(n - k) - lfact(k)) end function Base.factorial{T<:FloatingPoint}(n::T, k::T) exp(lfact(n) - lfact(k)) end ⊞{T<:Real}(n::T, k::T) = binomial(n, k) ⊠{T<:Real}(n::T, k::T) = factorial(n, n-k) </syntaxhighlight> '''Main''' <syntaxhighlight lang="julia"> function picknk{T<:Integer}(lo::T, hi::T) n = rand(lo:hi) k = rand(1:n) return (n, k) end nsamp = 10 print("Tests of the combinations (⊞) and permutations (⊠) operators for ") println("integer values.") println() lo, hi = 1, 12 print(nsamp, " samples of n ⊠ k with n in [", lo, ", ", hi, "] ") println("and k in [1, n].") for i in 1:nsamp (n, k) = picknk(lo, hi) println(@sprintf " %2d ⊠ %2d = %18d" n k n ⊠ k) end lo, hi = 10, 60 println() print(nsamp, " samples of n ⊞ k with n in [", lo, ", ", hi, "] ") println("and k in [1, n].") for i in 1:nsamp (n, k) = picknk(lo, hi) println(@sprintf " %2d ⊞ %2d = %18d" n k n ⊞ k) end println() print("Tests of the combinations (⊞) and permutations (⊠) operators for ") println("(big) float values.") println() lo, hi = 5, 15000 print(nsamp, " samples of n ⊠ k with n in [", lo, ", ", hi, "] ") println("and k in [1, n].") for i in 1:nsamp (n, k) = picknk(lo, hi) n = BigFloat(n) k = BigFloat(k) println(@sprintf " %7.1f ⊠ %7.1f = %10.6e" n k n ⊠ k) end lo, hi = 100, 1000 println() print(nsamp, " samples of n ⊞ k with n in [", lo, ", ", hi, "] ") println("and k in [1, n].") for i in 1:nsamp (n, k) = picknk(lo, hi) n = BigFloat(n) k = BigFloat(k) println(@sprintf " %7.1f ⊞ %7.1f = %10.6e" n k n ⊞ k) end </syntaxhighlight> {{out}} <pre> Tests of the combinations (⊞) and permutations (⊠) operators for integer values. 10 samples of n ⊠ k with n in [1, 12] and k in [1, n]. 4 ⊠ 2 = 12 9 ⊠ 2 = 72 2 ⊠ 1 = 2 8 ⊠ 2 = 56 7 ⊠ 5 = 2520 4 ⊠ 2 = 12 9 ⊠ 8 = 362880 11 ⊠ 6 = 332640 1 ⊠ 1 = 1 8 ⊠ 5 = 6720 10 samples of n ⊞ k with n in [10, 60] and k in [1, n]. 58 ⊞ 26 = 22150361247847371 22 ⊞ 21 = 22 32 ⊞ 30 = 496 11 ⊞ 4 = 330 32 ⊞ 29 = 4960 16 ⊞ 7 = 11440 31 ⊞ 25 = 736281 13 ⊞ 13 = 1 43 ⊞ 28 = 151532656696 49 ⊞ 37 = 92263734836 Tests of the combinations (⊞) and permutations (⊠) operators for (big) float values. 10 samples of n ⊠ k with n in [5, 15000] and k in [1, n]. 8375.0 ⊠ 5578.0 = 2.200496e+20792 1556.0 ⊠ 592.0 = 1.313059e+1833 1234.0 ⊠ 910.0 = 2.231762e+2606 12531.0 ⊠ 9361.0 = 2.339542e+36188 12418.0 ⊠ 6119.0 = 1.049662e+24251 9435.0 ⊠ 8960.0 = 4.273644e+32339 9430.0 ⊠ 5385.0 = 1.471741e+20551 9876.0 ⊠ 5386.0 = 9.073417e+20712 941.0 ⊠ 911.0 = 8.689430e+2358 8145.0 ⊠ 4357.0 = 1.368129e+16407 10 samples of n ⊞ k with n in [100, 1000] and k in [1, n]. 757.0 ⊞ 237.0 = 6.813837e+202 457.0 ⊞ 413.0 = 4.816707e+61 493.0 ⊞ 372.0 = 8.607443e+117 206.0 ⊞ 26.0 = 6.911828e+32 432.0 ⊞ 300.0 = 1.248351e+114 650.0 ⊞ 469.0 = 3.284854e+165 203.0 ⊞ 115.0 = 1.198089e+59 583.0 ⊞ 429.0 = 5.700279e+144 329.0 ⊞ 34.0 = 2.225630e+46 464.0 ⊞ 178.0 = 5.615925e+132 </pre> =={{header\|Kotlin}}== As Kotlin/JVM can use the java.math.BigInteger class, there is no need to use floating point approximations and so we use exact integer arithmetic for all parts of the task. <syntaxhighlight lang="scala">// version 1.1.2 import java.math.BigInteger fun perm(n: Int, k: Int): BigInteger { require(n > 0 && k >= 0) return (n - k + 1 .. n).fold(BigInteger.ONE) { acc, i -> acc * BigInteger.valueOf(i.toLong()) } } fun comb(n: Int, k: Int): BigInteger { require(n > 0 && k >= 0) val fact = (2..k).fold(BigInteger.ONE) { acc, i -> acc * BigInteger.valueOf(i.toLong()) } return perm(n, k) / fact } fun main(args: Array<String>) { println("A sample of permutations from 1 to 12:") for (n in 1..12) System.out.printf("%2d P %-2d = %d\n", n, n / 3, perm(n, n / 3)) println("\nA sample of combinations from 10 to 60:") for (n in 10..60 step 10) System.out.printf("%2d C %-2d = %d\n", n, n / 3, comb(n, n / 3)) println("\nA sample of permutations from 5 to 15000:") val na = intArrayOf(5, 50, 500, 1000, 5000, 15000) for (n in na) { val k = n / 3 val s = perm(n, k).toString() val l = s.length val e = if (l <= 40) "" else "... (${l - 40} more digits)" System.out.printf("%5d P %-4d = %s%s\n", n, k, s.take(40), e) } println("\nA sample of combinations from 100 to 1000:") for (n in 100..1000 step 100) { val k = n / 3 val s = comb(n, k).toString() val l = s.length val e = if (l <= 40) "" else "... (${l - 40} more digits)" System.out.printf("%4d C %-3d = %s%s\n", n, k, s.take(40), e) } }</syntaxhighlight> {{out}} <pre> A sample of permutations from 1 to 12: 1 P 0 = 1 2 P 0 = 1 3 P 1 = 3 4 P 1 = 4 5 P 1 = 5 6 P 2 = 30 7 P 2 = 42 8 P 2 = 56 9 P 3 = 504 10 P 3 = 720 11 P 3 = 990 12 P 4 = 11880 A sample of combinations from 10 to 60: 10 C 3 = 120 20 C 6 = 38760 30 C 10 = 30045015 40 C 13 = 12033222880 50 C 16 = 4923689695575 60 C 20 = 4191844505805495 A sample of permutations from 5 to 15000: 5 P 1 = 5 50 P 16 = 103017324974226408345600000 500 P 166 = 3534874921742942787609361826601762306844... (395 more digits) 1000 P 333 = 5969326288503415089039701765900784280998... (932 more digits) 5000 P 1666 = 6856745757255674275484536940248896062234... (5986 more digits) 15000 P 5000 = 9649853988727493922014858805931295980792... (20430 more digits) A sample of combinations from 100 to 1000: 100 C 33 = 294692427022540894366527900 200 C 66 = 7269752545169278341527066651192738976755... (14 more digits) 300 C 100 = 4158251463258564744783383526326405580280... (42 more digits) 400 C 133 = 1257948684182108702133348475651965004491... (70 more digits) 500 C 166 = 3926028386194422755220408345072331428197... (97 more digits) 600 C 200 = 2506017783221402805005616770513228835202... (125 more digits) 700 C 233 = 8103203563339599904740453644031138232944... (152 more digits) 800 C 266 = 2645623362683627034288829299556124255091... (180 more digits) 900 C 300 = 1743356373296446642960730765085718347630... (208 more digits) 1000 C 333 = 5776134553147651669777486323549601722339... (235 more digits) </pre> =={{header\|Lua}}== {{trans\|Perl}} <syntaxhighlight lang="Lua"> -- Helper function to display results in scientific notation corrected function eshow(x) local e = math.floor(x / math.log(10)) local exponentiated = math.exp(x - e * math.log(10)) -- Corrected format specifier from %.8Fe%+d to %.8e, and manually constructing the scientific notation if needed return string.format("%.8e", exponentiated) .. "e" .. tostring(e) end -- The rest of the functions remain the same -- Define the factorial function for permutations function P(n, k) local x = 1 for i = n - k + 1, n do x = x * i end return x end -- Define the function for big permutations using logarithms function P_big(n, k) local x = 0 for i = n - k + 1, n do x = x + math.log(i) end return eshow(x) end -- Define the combinations function function C(n, k) local x = 1 for i = 1, k do x = x * (n - i + 1) / i end return x end -- Define the function for big combinations using logarithms function C_big(n, k) local x = 0 for i = 1, k do x = x + math.log(n - i + 1) - math.log(i) end return math.exp(x) end -- Function to showcase the calculations function showoff(text, code, fname, ...) local n = {...} print("\nA sample of " .. text .. " from " .. n[1] .. " to " .. n[#n] .. "") for _, v in ipairs(n) do local k = math.floor(v / 3) print(v, fname .. " " .. k .. " = ", code(v, k)) end end -- Examples of usage showoff("Permutations", P, "P", 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) showoff("Combinations", C, "C", 10, 20, 30, 40, 50, 60) showoff("Permutations", P_big, "P", 5, 50, 500, 1000, 5000, 15000) showoff("Combinations", C_big, "C", 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000) </syntaxhighlight> {{out}} <pre> A sample of Permutations from 1 to 12 1 P 0 = 1 2 P 0 = 1 3 P 1 = 3 4 P 1 = 4 5 P 1 = 5 6 P 2 = 30 7 P 2 = 42 8 P 2 = 56 9 P 3 = 504 10 P 3 = 720 11 P 3 = 990 12 P 4 = 11880 A sample of Combinations from 10 to 60 10 C 3 = 120.0 20 C 6 = 38760.0 30 C 10 = 30045015.0 40 C 13 = 12033222880.0 50 C 16 = 4923689695575.0 60 C 20 = 4.1918445058055e+15 A sample of Permutations from 5 to 15000 5 P 1 = 5.00000000e+00e0 50 P 16 = 1.03017325e+00e26 500 P 166 = 3.53487492e+00e434 1000 P 333 = 5.96932629e+00e971 5000 P 1666 = 6.85674576e+00e6025 15000 P 5000 = 9.64985399e+00e20469 A sample of Combinations from 100 to 1000 100 C 33 = 2.9469242702254e+26 200 C 66 = 7.2697525451696e+53 300 C 100 = 4.158251463258e+81 400 C 133 = 1.2579486841822e+109 500 C 166 = 3.9260283861944e+136 600 C 200 = 2.5060177832203e+164 700 C 233 = 8.1032035633383e+191 800 C 266 = 2.6456233626831e+219 900 C 300 = 1.7433563732974e+247 1000 C 333 = 5.776134553149e+274 </pre> =={{header\|M2000 Interpreter}}== <syntaxhighlight lang="m2000 interpreter"> Module PermComb { Form 80, 50 perm=lambda (x,y) ->{ def i,z z=1 For i=x-y+1 to x :z=i:next i =z } fact=lambda (x) ->{ def i,z z=1 For i=2 to x :z=i:next i =z } comb=lambda (x as decimal, y as decimal) ->{ If y>x then { =0 } else.if x=y then { =1 } else { if x-y<y then y=x-y def decimal i, z=1, ym ym=y For i=x to x-y+1 z=i z=z/ym ym-- : if ym<1 then ym=1@ next i =round(z,0) } } Document Doc$ WriteLn("-- Permutations - from 1 to 12") For i=1 to 12 l$="" : For j=1 to i : l$+= format$("P({0},{1})={2} ",i, j,perm(i, j)) :next j Writetext(l$) next i WriteLn("-- Combinations from 10 to 60") For i=10 to 60 step 10 l$="" : For j=1 to i step i div 5 : l$+= format$("C({0},{1})={2} ",i, j,comb(i, j)) :next j Writetext(l$) Next i WriteLn("-- Permutations from 5000 to 15000") For i=5000 to 15000 step 5000 l$="" : For j=10 to 70 step 20: l$+= format$("P({0},{1})={2} ",i, j,perm(i, j)) :next j Writetext(l$) Next i WriteLn("-- Combinations from 200 to 1000") For i=200 to 1000 step 200 l$="" : For j=20 to 100 step 20: l$+= format$("C({0},{1})={2} ",i, j,comb(i, j)) :next j Writetext(l$) Next i ClipBoard Doc$ Sub WriteText(a$) doc$=a$+{ } Report a$ End Sub Sub WriteLn(a$) doc$=a$+{ } Print a$ End Sub } PermComb </syntaxhighlight> {{out}} -- Permutations - from 1 to 12 P(1,1)=1 P(2,1)=2 P(2,2)=2 P(3,1)=3 P(3,2)=6 P(3,3)=6 P(4,1)=4 P(4,2)=12 P(4,3)=24 P(4,4)=24 P(5,1)=5 P(5,2)=20 P(5,3)=60 P(5,4)=120 P(5,5)=120 P(6,1)=6 P(6,2)=30 P(6,3)=120 P(6,4)=360 P(6,5)=720 P(6,6)=720 P(7,1)=7 P(7,2)=42 P(7,3)=210 P(7,4)=840 P(7,5)=2520 P(7,6)=5040 P(7,7)=5040 P(8,1)=8 P(8,2)=56 P(8,3)=336 P(8,4)=1680 P(8,5)=6720 P(8,6)=20160 P(8,7)=40320 P(8,8)=40320 P(9,1)=9 P(9,2)=72 P(9,3)=504 P(9,4)=3024 P(9,5)=15120 P(9,6)=60480 P(9,7)=181440 P(9,8)=362880 P(9,9)=362880 P(10,1)=10 P(10,2)=90 P(10,3)=720 P(10,4)=5040 P(10,5)=30240 P(10,6)=151200 P(10,7)=604800 P(10,8)=1814400 P(10,9)=3628800 P(10,10)=3628800 P(11,1)=11 P(11,2)=110 P(11,3)=990 P(11,4)=7920 P(11,5)=55440 P(11,6)=332640 P(11,7)=1663200 P(11,8)=6652800 P(11,9)=19958400 P(11,10)=39916800 P(11,11)=39916800 P(12,1)=12 P(12,2)=132 P(12,3)=1320 P(12,4)=11880 P(12,5)=95040 P(12,6)=665280 P(12,7)=3991680 P(12,8)=19958400 P(12,9)=79833600 P(12,10)=239500800 P(12,11)=479001600 P(12,12)=479001600 -- Combinations from 10 to 60 C(10,1)=10 C(10,3)=120 C(10,5)=252 C(10,7)=120 C(10,9)=10 C(20,1)=20 C(20,5)=15504 C(20,9)=167960 C(20,13)=77520 C(20,17)=1140 C(30,1)=30 C(30,7)=2035800 C(30,13)=119759850 C(30,19)=54627300 C(30,25)=142506 C(40,1)=40 C(40,9)=273438880 C(40,17)=88732378800 C(40,25)=40225345056 C(40,33)=18643560 C(50,1)=50 C(50,11)=37353738800 C(50,21)=67327446062800 C(50,31)=30405943383200 C(50,41)=2505433700 C(60,1)=60 C(60,13)=5166863427600 C(60,25)=51915437974328292 C(60,37)=23385332420868600 C(60,49)=342700125300 -- Permutations from 5000 to 15000 P(5000,10)=9,67807348145655E+36 P(5000,30)=8,53575581200676E+110 P(5000,50)=6,94616656703754E+184 P(5000,70)=5,21383580146195E+258 P(10000,10)=9,95508690556325E+39 P(10000,30)=9,57391540294832E+119 P(10000,50)=8,84526658067387E+199 P(10000,70)=7,850079552152E+279 P(15000,10)=5,74922667554068E+41 P(15000,30)=1,86266591363916E+125 P(15000,50)=5,87565776023335E+208 P(15000,70)=1,80450662858719E+292 -- Combinations from 200 to 1000 C(200,20)=1613587787967350073386147640 C(200,40)=721126811024990370 C(200,60)=286107190317772000463240955 C(200,80)=900482 C(200,100)=478205104 C(400,20)=3558235073 C(400,40)=130321 C(400,60)=74780600187861332802765 C(400,80)=9521248771125 C(400,100)=447355513982663594791392 C(600,20)=2801445153584 C(600,40)=266319106596345 C(600,60)=14409368913 C(600,80)=271441 C(600,100)=52868467287780595308 C(800,20)=1925279023672620 C(800,40)=23121069591511231041 C(800,60)=18702067923763447158 C(800,80)=1193559552292625 C(800,100)=172727802 C(1000,20)=1239329180287869852 C(1000,40)=1937726921514640866484017 C(1000,60)=149470629867337347460963500 C(1000,80)=1269150275532146867313740 C(1000,100)=10088410532027029794548 =={{header\|Maple}}== <syntaxhighlight lang="maple"> comb := proc (n::integer, k::integer) return factorial(n)/(factorial(k)factorial(n-k)); end proc; perm := proc (n::integer, k::integer) return factorial(n)/factorial(n-k); end proc; </syntaxhighlight> =={{header\|Mathematica}} / {{header\|Wolfram Language}}== <syntaxhighlight lang="mathematica">ClearAll[Combination,Permutation] Combination[n_,k_]:=Binomial[n,k] Permutation[n_,k_]:=Binomial[n,k]k! TableForm[Array[Permutation,{12,12}],TableHeadings->{Range[12],Range[12]}] TableForm[Array[Combination,{6,6},{{10,60},{10,60}}],TableHeadings->{Range[10,60,10],Range[10,60,10]}] {Row[{#,"P",#-2}," "],N@Permutation[#,#-2]}&/@{5,1000,5000,10000,15000}//Grid {Row[{#,"C",#/2}," "],N@Combination[#,#/2]}&/@Range[100,1000,100]//Grid</syntaxhighlight> {{out}} <pre> 1 2 3 4 5 6 7 8 9 10 11 12 1 1 0 0 0 0 0 0 0 0 0 0 0 2 2 2 0 0 0 0 0 0 0 0 0 0 3 3 6 6 0 0 0 0 0 0 0 0 0 4 4 12 24 24 0 0 0 0 0 0 0 0 5 5 20 60 120 120 0 0 0 0 0 0 0 6 6 30 120 360 720 720 0 0 0 0 0 0 7 7 42 210 840 2520 5040 5040 0 0 0 0 0 8 8 56 336 1680 6720 20160 40320 40320 0 0 0 0 9 9 72 504 3024 15120 60480 181440 362880 362880 0 0 0 10 10 90 720 5040 30240 151200 604800 1814400 3628800 3628800 0 0 11 11 110 990 7920 55440 332640 1663200 6652800 19958400 39916800 39916800 0 12 12 132 1320 11880 95040 665280 3991680 19958400 79833600 239500800 479001600 479001600 10 20 30 40 50 60 10 1 0 0 0 0 0 20 184756 1 0 0 0 0 30 30045015 30045015 1 0 0 0 40 847660528 137846528820 847660528 1 0 0 50 10272278170 47129212243960 47129212243960 10272278170 1 0 60 75394027566 4191844505805495 118264581564861424 4191844505805495 75394027566 1 5 P 3 60. 1000 P 998 2.01193630038546910^2567 5000 P 4998 2.11428896330277210^16325 10000 P 9998 1.42312984045852710^35659 15000 P 14998 1.37329951674258410^56129 100 C 50 1.0089110^29 200 C 100 9.0548510^58 300 C 150 9.3759710^88 400 C 200 1.0295310^119 500 C 250 1.1674410^149 600 C 300 1.3510810^179 700 C 350 1.5857410^209 800 C 400 1.8804210^239 900 C 450 2.2474710^269 1000 C 500 2.7028810^299 </pre> Note that Mathematica can easily handle very big numbers with exact integer arithmetic: <syntaxhighlight lang="mathematica"> Permutation[200000, 100000] </syntaxhighlight> {{out}} The output is 516777 digits longs: <pre> 50287180689616781338617355322585606........0321815299686400000000000000000000......(lots of zeroes) </pre> =={{header\|МК-61/52}}== <syntaxhighlight lang="text">П2 <-> П1 -> <-> П7 КПП7 С/П ИП1 ИП2 - ПП 53 П3 ИП1 ПП 53 ИП3 / В/О 1 ИП1 * L2 21 В/О ИП1 ИП2 - ПП 53 П3 ИП2 ПП 53 ИП3 * П3 ИП1 ПП 53 ИП3 / В/О ИП1 ИП2 + 1 - П1 ПП 26 В/О ВП П0 1 ИП0 * L0 56 В/О</syntaxhighlight> ''Input'': ''x ^ n ^ k В/О С/П'', where ''x'' = 8 for permutations; 20 for permutations with repetitions; 26 for combinations; 44 for combinations with repetitions. Printing of test cases is performed incrementally, which is associated with the characteristics of the device output. =={{header\|Nim}}== <syntaxhighlight lang="nim">import bigints proc perm(n, k: int32): BigInt = result = initBigInt 1 var k = n - k n = n while n > k: result = n dec n proc comb(n, k: int32): BigInt = result = perm(n, k) var k = k while k > 0: result = result div k dec k echo "P(1000, 969) = ", perm(1000, 969) echo "C(1000, 969) = ", comb(1000, 969)</syntaxhighlight> =={{header\|PARI/GP}}== <syntaxhighlight lang="parigp">sample(f,a,b)=for(i=1,4, my(n1=random(b-a)+a,n2=random(b-a)+a); [n1,n2]=[max(n1,n2),min(n1,n2)]; print(n1", "n2": "f(n1,n2))) permExact(m,n)=factorback([m-n+1..m]); combExact=binomial; permApprox(m,n)=exp(lngamma(m+1)-lngamma(m-n+1)); combApprox(m,n)=exp(lngamma(m+1)-lngamma(n+1)-lngamma(m-n+1)); sample(permExact, 1, 12); sample(combExact, 10, 60); sample(permApprox, 5, 15000); sample(combApprox, 100, 1000);</syntaxhighlight> {{out}} <pre>?sample(permExact, 1, 12); 8, 2: 56 11, 8: 6652800 9, 9: 362880 6, 1: 6 ?sample(combExact, 10, 60); 46, 14: 239877544005 34, 22: 548354040 51, 20: 77535155627160 49, 26: 58343356817424 ?sample(permApprox, 5, 15000); 8374, 8306: 6.6786635386843773562533982329356314192 E29119 4064, 2497: 7.7325589445068984950461595444827041944 E8575 13234, 784: 1.3439405881429921844444755481930625437 E3221 14136, 1523: 9.7219281356264565060667995087812528666 E6283 ?sample(combApprox, 100, 1000); 988, 702: 4.1430346142101709187524161097370204275 E256 861, 225: 1.9423942269910057792279495652023745087 E213 580, 350: 4.9721729266474994835623000459244303642 E167 977, 846: 6.0586575447000334467351859308510379521 E165</pre> =={{header\|Perl}}== Although Perl can handle arbitrarily large numbers using Math::BigInt and Math::BigFloat, it's native integers and floats are limited to what the computer's native types can handle. As with the Raku code, some special handling was done for those values which would have overflowed the native floating point type. <syntaxhighlight lang="perl">use strict; use warnings; showoff( "Permutations", \&P, "P", 1 .. 12 ); showoff( "Combinations", \&C, "C", map $_10, 1..6 ); showoff( "Permutations", \&P_big, "P", 5, 50, 500, 1000, 5000, 15000 ); showoff( "Combinations", \&C_big, "C", map $_100, 1..10 ); sub showoff { my ($text, $code, $fname, @n) = @_; print "\nA sample of $text from $n[0] to $n[-1]\n"; for my $n ( @n ) { my $k = int( $n / 3 ); print $n, " $fname $k = ", $code->($n, $k), "\n"; } } sub P { my ($n, $k) = @_; my $x = 1; $x = $_ for $n - $k + 1 .. $n ; $x; } sub P_big { my ($n, $k) = @_; my $x = 0; $x += log($_) for $n - $k + 1 .. $n ; eshow($x); } sub C { my ($n, $k) = @_; my $x = 1; $x = ($n - $_ + 1) / $_ for 1 .. $k; $x; } sub C_big { my ($n, $k) = @_; my $x = 0; $x += log($n - $_ + 1) - log($_) for 1 .. $k; exp($x); } sub eshow { my ($x) = @_; my $e = int( $x / log(10) ); sprintf "%.8Fe%+d", exp($x - $e log(10)), $e; } </syntaxhighlight> {{out}} <pre style="height:50ex">A sample of Permutations from 1 to 12 1 P 0 = 1 2 P 0 = 1 3 P 1 = 3 4 P 1 = 4 5 P 1 = 5 6 P 2 = 30 7 P 2 = 42 8 P 2 = 56 9 P 3 = 504 10 P 3 = 720 11 P 3 = 990 12 P 4 = 11880 A sample of Combinations from 10 to 60 10 C 3 = 120 20 C 6 = 38760 30 C 10 = 30045015 40 C 13 = 12033222880 50 C 16 = 4923689695575 60 C 20 = 4.1918445058055e+15 A sample of Permutations from 5 to 15000 5 P 1 = 5.00000000e+0 50 P 16 = 1.03017325e+26 500 P 166 = 3.53487492e+434 1000 P 333 = 5.96932629e+971 5000 P 1666 = 6.85674576e+6025 15000 P 5000 = 9.64985399e+20469 A sample of Combinations from 100 to 1000 100 C 33 = 2.94692427022544e+26 200 C 66 = 7.26975254516929e+53 300 C 100 = 4.15825146325788e+81 400 C 133 = 1.25794868418216e+109 500 C 166 = 3.92602838619369e+136 600 C 200 = 2.50601778322159e+164 700 C 233 = 8.10320356333741e+191 800 C 266 = 2.64562336268385e+219 900 C 300 = 1.74335637329697e+247 1000 C 333 = 5.77613455314442e+274</pre> =={{header\|Phix}}== Translation of Raku/Sidef, same results<br> Update: there are now builtin routines k_perm(n,k) and choose(n,k), slightly more efficient equivalents of P() and C() respectively. <!--<syntaxhighlight lang="phix">(phixonline)--> <span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span> <span style="color: #008080;">function</span> <span style="color: #000000;">P</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #7060A8;">factorial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)/</span><span style="color: #7060A8;">factorial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span> <span style="color: #008080;">function</span> <span style="color: #000000;">C</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #000000;">P</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)/</span><span style="color: #7060A8;">factorial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span> <span style="color: #008080;">function</span> <span style="color: #000000;">lstirling</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;"><</span><span style="color: #000000;">10</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">lstirling</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)-</span><span style="color: #7060A8;">log</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #008080;">return</span> <span style="color: #000000;">0.5</span><span style="color: #0000FF;"></span><span style="color: #7060A8;">log</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;"></span><span style="color: #004600;">PI</span><span style="color: #0000FF;"></span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">n</span><span style="color: #0000FF;"></span><span style="color: #7060A8;">log</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #004600;">E</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">/(</span><span style="color: #000000;">12</span><span style="color: #0000FF;"></span><span style="color: #004600;">E</span><span style="color: #0000FF;"></span><span style="color: #000000;">n</span><span style="color: #0000FF;">))</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span> <span style="color: #008080;">function</span> <span style="color: #000000;">P_approx</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #000000;">lstirling</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">lstirling</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span> <span style="color: #008080;">function</span> <span style="color: #000000;">C_approx</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #000000;">lstirling</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">lstirling</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">lstirling</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span> <span style="color: #008080;">function</span> <span style="color: #000000;">to_s</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">v</span><span style="color: #0000FF;">/</span><span style="color: #7060A8;">log</span><span style="color: #0000FF;">(</span><span style="color: #000000;">10</span><span style="color: #0000FF;">))</span> <span style="color: #008080;">return</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%.9ge%d"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #004600;">E</span><span style="color: #0000FF;">,</span><span style="color: #000000;">v</span><span style="color: #0000FF;">-</span><span style="color: #000000;">e</span><span style="color: #0000FF;"></span><span style="color: #7060A8;">log</span><span style="color: #0000FF;">(</span><span style="color: #000000;">10</span><span style="color: #0000FF;">)),</span><span style="color: #000000;">e</span><span style="color: #0000FF;">})</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span> <span style="color: #000080;font-style:italic;">-- Test code</span> <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"=> Exact results:\n"</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">for</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">12</span> <span style="color: #008080;">do</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"P(%d,%d) = %d\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">P</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)})</span> <span style="color: #008080;">end</span> <span style="color: #008080;">for</span> <span style="color: #008080;">for</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">10</span> <span style="color: #008080;">to</span> <span style="color: #000000;">60</span> <span style="color: #008080;">by</span> <span style="color: #000000;">10</span> <span style="color: #008080;">do</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"C(%d,%d) = %d\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">C</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)})</span> <span style="color: #008080;">end</span> <span style="color: #008080;">for</span> <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"=> Floating point approximations:\n"</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">constant</span> <span style="color: #000000;">tests</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">5</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">50</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">500</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1000</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">5000</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">15000</span><span style="color: #0000FF;">}</span> <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tests</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">tests</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"P(%d,%d) = %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">to_s</span><span style="color: #0000FF;">(</span><span style="color: #000000;">P_approx</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">))})</span> <span style="color: #008080;">end</span> <span style="color: #008080;">for</span> <span style="color: #008080;">for</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">100</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1000</span> <span style="color: #008080;">by</span> <span style="color: #000000;">100</span> <span style="color: #008080;">do</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"C(%d,%d) = %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">to_s</span><span style="color: #0000FF;">(</span><span style="color: #000000;">C_approx</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">))})</span> <span style="color: #008080;">end</span> <span style="color: #008080;">for</span> <!--</syntaxhighlight>--> {{out}} <pre> => Exact results: P(1,0) = 1 P(2,0) = 1 P(3,1) = 3 P(4,1) = 4 P(5,1) = 5 P(6,2) = 30 P(7,2) = 42 P(8,2) = 56 P(9,3) = 504 P(10,3) = 720 P(11,3) = 990 P(12,4) = 11880 C(10,3) = 120 C(20,6) = 38760 C(30,10) = 30045014 C(40,13) = 12033222880 C(50,16) = 4923689695575 C(60,20) = 4191844505805496 => Floating point approximations: P(5,1) = 5e0 P(50,16) = 1.03017326e26 P(500,166) = 3.53487492e434 P(1000,333) = 5.96932629e971 P(5000,1666) = 6.85674576e6025 P(15000,5000) = 9.64985399e20469 C(100,33) = 2.94692433e26 C(200,66) = 7.26975256e53 C(300,100) = 4.15825147e81 C(400,133) = 1.25794868e109 C(500,166) = 3.92602839e136 C(600,200) = 2.50601778e164 C(700,233) = 8.10320356e191 C(800,266) = 2.64562336e219 C(900,300) = 1.74335637e247 C(1000,333) = 5.77613455e274 </pre> =={{header\|Python}}== ==={{libheader\|SciPy}}=== <syntaxhighlight lang="python">from __future__ import print_function from scipy.misc import factorial as fact from scipy.misc import comb def perm(N, k, exact=0): return comb(N, k, exact) fact(k, exact) exact=True print('Sample Perms 1..12') for N in range(1, 13): k = max(N-2, 1) print('%iP%i =' % (N, k), perm(N, k, exact), end=', ' if N % 5 else '\n') print('\n\nSample Combs 10..60') for N in range(10, 61, 10): k = N-2 print('%iC%i =' % (N, k), comb(N, k, exact), end=', ' if N % 50 else '\n') exact=False print('\n\nSample Perms 5..1500 Using FP approximations') for N in [5, 15, 150, 1500, 15000]: k = N-2 print('%iP%i =' % (N, k), perm(N, k, exact)) print('\nSample Combs 100..1000 Using FP approximations') for N in range(100, 1001, 100): k = N-2 print('%iC%i =' % (N, k), comb(N, k, exact)) </syntaxhighlight> {{out}} <pre>Sample Perms 1..12 1P1 = 1, 2P1 = 2, 3P1 = 3, 4P2 = 12, 5P3 = 60 6P4 = 360, 7P5 = 2520, 8P6 = 20160, 9P7 = 181440, 10P8 = 1814400 11P9 = 19958400, 12P10 = 239500800, Sample Combs 10..60 10C8 = 45, 20C18 = 190, 30C28 = 435, 40C38 = 780, 50C48 = 1225 60C58 = 1770, Sample Perms 5..1500 Using FP approximations 5P3 = 60.0 15P13 = 653837184000.0 150P148 = 2.85669197822e+262 1500P1498 = inf 15000P14998 = inf Sample Combs 100..1000 Using FP approximations 100C98 = 4950.0 200C198 = 19900.0 300C298 = 44850.0 400C398 = 79800.0 500C498 = 124750.0 600C598 = 179700.0 700C698 = 244650.0 800C798 = 319600.0 900C898 = 404550.0 1000C998 = 499500.0</pre> =={{header\|R}}== R has most of this built in. The documentation for choose warns that it only calculates its result directly if k is small. Skimming [https://github.com/wch/r-source/blob/trunk/src/nmath/choose.c what appears to be the source code] suggests that 29 is the highest "small" k. This means that we can solve both tasks with little more than R's choose. <syntaxhighlight lang="rsplus">perm <- function(n, k) choose(n, k) * factorial(k) print(perm(seq(from = 3, to = 12, by = 3), seq(from = 2, to = 8, by = 2))) print(choose(seq(from = 10, to = 60, by = 10), seq(from = 3, to = 18, by = 3))) print(perm(seq(from = 1500, to = 15000, by = 1500), seq(from = 55, to = 100, by = 5))) print(choose(seq(from = 100, to = 1000, by = 150), seq(from = 70, to = 100, by = 5)))</syntaxhighlight> {{out}} <pre>> print(perm(seq(from = 3, to = 12, by = 3), seq(from = 2, to = 8, by = 2))) [1] 6 360 60480 19958400 > print(choose(seq(from = 10, to = 60, by = 10), seq(from = 3, to = 18, by = 3))) [1] 1.200000e+02 3.876000e+04 1.430715e+07 5.586853e+09 2.250830e+12 9.250296e+14 > print(perm(seq(from = 1500, to = 15000, by = 1500), seq(from = 55, to = 100, by = 5))) [1] 1.777313e+174 2.340618e+208 1.807619e+237 1.972900e+264 2.940161e+290 Inf Inf Inf Inf Inf > print(choose(seq(from = 100, to = 1000, by = 150), seq(from = 70, to = 100, by = 5))) [1] 2.937234e+25 1.158870e+65 4.228142e+85 3.202323e+101 1.944760e+115 8.124447e+127 6.385051e+139</pre> We can notice some interesting things about R from this task. Despite being a mathematical programming language: #R does not have any function like perm built in. #R does not have big integers by default and will return Inf without warning. #R's documentation does not tell us exactly when choose's behaviour changes. =={{header\|Racket}}== Racket's "math" library has two functions that compute nCk and nPk. They work only on integers, but since Racket supports unlimited integers there is no need for a floating point estimate: <syntaxhighlight lang="racket"> #lang racket (require math) (define C binomial) (define P permutations) (C 1000 10) ; -> 263409560461970212832400 (P 1000 10) ; -> 955860613004397508326213120000 </syntaxhighlight> (I'll spare this page from yet another big listing of samples...) =={{header\|Raku}}== (formerly Perl 6) Raku can't compute arbitrary large floating point values, thus we will use logarithms, as is often needed when dealing with combinations. We'll also use a Stirling method to approximate <math>\ln(n!)</math>: <math>\ln n! \approx \frac{1}{2}\ln(2\pi n) + n\ln\left(\frac{n}{e} + \frac{1}{12 e n}\right)</math> Notice that Raku can process arbitrary long integers, though. So it's not clear whether using floating points is useful in this case. <syntaxhighlight lang="raku" line>multi P($n, $k) { [] $n - $k + 1 .. $n } multi C($n, $k) { P($n, $k) / [] 1 .. $k } sub lstirling(\n) { n < 10 ?? lstirling(n+1) - log(n+1) !! .5log(2pin)+ nlog(n/e+1/(12en)) } role Logarithm { method gist { my $e = (self/10.log).Int; sprintf "%.8fE%+d", exp(self - $e10.log), $e; } } multi P($n, $k, :$float!) { (lstirling($n) - lstirling($n -$k)) but Logarithm } multi C($n, $k, :$float!) { (lstirling($n) - lstirling($n -$k) - lstirling($k)) but Logarithm } say "Exact results:"; for 1..12 -> $n { my $p = $n div 3; say "P($n, $p) = ", P($n, $p); } for 10, 20 ... 60 -> $n { my $p = $n div 3; say "C($n, $p) = ", C($n, $p); } say ''; say "Floating point approximations:"; for 5, 50, 500, 1000, 5000, 15000 -> $n { my $p = $n div 3; say "P($n, $p) = ", P($n, $p, :float); } for 100, 200 ... 1000 -> $n { my $p = $n div 3; say "C($n, $p) = ", C($n, $p, :float); }</syntaxhighlight> {{out}} <pre>Exact results: P(1, 0) = 1 P(2, 0) = 1 P(3, 1) = 3 P(4, 1) = 4 P(5, 1) = 5 P(6, 2) = 30 P(7, 2) = 42 P(8, 2) = 56 P(9, 3) = 504 P(10, 3) = 720 P(11, 3) = 990 P(12, 4) = 11880 C(10, 3) = 120 C(20, 6) = 38760 C(30, 10) = 30045015 C(40, 13) = 12033222880 C(50, 16) = 4923689695575 C(60, 20) = 4191844505805495 Floating point approximations: P(5, 1) = 5.00000000E+0 P(50, 16) = 1.03017326E+26 P(500, 166) = 3.53487492E+434 P(1000, 333) = 5.96932629E+971 P(5000, 1666) = 6.85674576E+6025 P(15000, 5000) = 9.64985399E+20469 C(100, 33) = 2.94692433E+26 C(200, 66) = 7.26975256E+53 C(300, 100) = 4.15825147E+81 C(400, 133) = 1.25794868E+109 C(500, 166) = 3.92602839E+136 C(600, 200) = 2.50601778E+164 C(700, 233) = 8.10320356E+191 C(800, 266) = 2.64562336E+219 C(900, 300) = 1.74335637E+247 C(1000, 333) = 5.77613455E+274</pre> =={{header\|REXX}}== The hard part of this REXX program was coding the   '''DO'''   loops for the various ranges. <~~lang~~syntaxhighlight lang="rexx">/REXX program to compute and displays a sampling of combinations and permutations. / numeric digits 100 /use ~~hundred~~100 decimal digits of precision. / do j=1 tofor 12; _= /show all permutations from ~~1──►~~ 121 ──► 12./ ~~_=;~~ do k=1 tofor j /step through all J permutations. / _=_ 'P('j","k')='perm(j,k)" " /add an extra blank between #snumbers. / end /k/ say strip(_) /show athe ~~horizontal~~permutations ~~line~~horizontally. of ~~PERMs~~/ end /j/ say /display a blank line for readability./ ~~say~~ do j=10 to 60 by 10; _= /show some combinations ~~10──►~~ 10 ──► 60. / ~~_=;~~ do k= 1 to j by j%5 /step through some combinations. / _=_ 'C('j","k')='comb(j,k)" " /add an extra blank between #snumbers. / end /k/ say strip(_) /show athe ~~horizontal~~combinations ~~line~~horizontally. of ~~COMBs~~/ end /j/ say /display a blank line for readability./ ~~say~~ numeric digits 20 /force floating point for big #snumbers./ do j=5 to 15000 by ~~1000~~ by 1000; _= /show a few permutations, big #snumbers./ ~~_=;~~ do k=1 to j for 5 by j%10 ~~for~~ 5 /gostep through some J permutations. / _=_ 'P('j","k')='perm(j,k)" " /add an extra blank between #snumbers. / end /k/ say strip(_) /show athe ~~horizontal~~permutations ~~line~~horizontally. of ~~PERMs~~/ end /j/ say /display a blank line for readability./ ~~say~~ do j=100 to 1000 by 100; _= /show a few combinations, big #snumbers./ ~~_=;~~ do k= 1 to j by j%5 /step through some combinations. / _=_ 'C('j","k')='comb(j,k)" " /add an extra blank between #snumbers. / end /k/ say strip(_) /show athe ~~horizontal~~combinations ~~line~~horizontally. of ~~COMBs~~/ end /j/ exit /stick a fork in it, we're all done. / /──────────────────────────────────────────────────────────────────────────────────────/ ~~/──────────────────────────────────COMB subroutine─────────────────────/~~ ~~comb~~perm: ~~procedure;~~ procedure; parse arg x,y; call ~~/args:~~.combPerm; X ~~things,~~ Y ~~at-a-time./~~ return _ if.combPerm: ~~y>x~~ ~~then~~ ~~return~~ 0 ~~/oops~~ _=1; do j=x-~~say,~~y+1 to ~~big~~x; a ~~chunk.~~ _=_j; end; / return _ if!: ~~x=y~~ ~~then~~ ~~return~~ procedure; parse arg x; !=1; do j=2 to x; /!=!j; X ~~things~~end; ~~same~~ as ~~chunk~~ ~~size.~~ return /! /──────────────────────────────────────────────────────────────────────────────────────/ ~~if x-y<y then y=x-y /switch things around for speed./~~ ~~call~~comb: ~~.cmbPrm~~ procedure; parse arg x,y /arguments: X things, ~~/call~~ ~~sub~~Y to ~~do heavy lifting~~at-a-time. / ~~return~~ ~~_/!(y)~~ if y >x then return 0 /~~perform~~oops-say, ~~one~~an ~~last division.~~error, too big a chunk./ if x =y then return 1 /X things are the same as chunk size./ ~~/──────────────────────────────────PERM subroutine─────────────────────/~~ ~~perm:~~ ~~procedure;~~ ~~parse~~ ~~arg~~ ~~x,y;~~ ~~call~~ ~~.cmbPrm;~~ if x-y <y then y=x - y /switch things around for speed. ~~return~~ _/ call .combPerm /call subroutine to do heavy lifting. / ~~/──────────────────────────────────.CMBPRM sugroutine──────────────────/~~ ~~.cmbPrm:~~ ~~_=1;~~ do ~~j=x-y+1~~ to x; return _~~=_j;~~ / !(y) ~~end;~~ /just perform one ~~return~~last _division. /</syntaxhighlight> ~~/──────────────────────────────────! subroutine────────────────────────/~~ ~~!: procedure; parse arg x; !=1; do j=2 to x; !=!j; end; return !</lang>~~ '''output''' <pre> ~~<pre style="overflow:scroll">~~ P(1,1)=1 P(2,1)=2 P(2,2)=2 Line 302 ⟶ 2,854: C(1000,1)=1000 C(1000,201)=2.6336937554862900107E+216 C(1000,401)=7.4293352412781479131E+290 C(1000,601)=3.3046738011675400053E+290 C(1000,801)=1.6522236106515115238E+215 </pre> =={{header\|RPL}}== ≪ / LAST ROT - ≫ '<span style="color:blue">BMOD</span>' STO ≪ → n k ≪ # 1d n k - 1 + n '''FOR''' j j * '''NEXT''' ≫ ≫ '<span style="color:blue">BPERM</span>' STO ≪ → n k ≪ # 1d DUP n k - 1 + n '''FOR''' j j * '''WHILE''' DUP2 SWAP <span style="color:blue">BMOD</span> # 0d == 3 PICK k R→B ≤ AND '''REPEAT''' OVER / SWAP 1 + SWAP '''END''' '''NEXT''' SWAP DROP ≫ ≫ '<span style="color:blue">BCOMB</span>' STO To mitigate risk of data overflow with 64-bits integers, the inner <code>WHILE..REPEAT</code> loop permanently attempts to divide the current numerator by increasing values of k. ≪ {} 1 11 '''FOR''' j 12 j <span style="color:blue">BPERM</span> + '''NEXT''' ≫ ≪ {} 10 50 '''FOR''' j 60 j <span style="color:blue">BCOMB</span> + 10 '''STEP''' ≫ {{out}} <pre> 2: { # 12d # 132d # 1320d # 11880d # 95040d # 665280d # 3991680d # 19958400d # 79833600d # 239500800d # 479001600d } 1: { # 75394027566d # 4191844505805495d # 118264581564861424d # 4191844505805495d # 75394027566d } </pre> Floating point approximations can be obtained by the built-in commands, provided that the result does not go beyond 1E500: 1000 150 PERM 1000 400 COMB {{out}} <pre> 2: 7.67405677335E444 1: 4.96527238625E290 </pre> =={{header\|Ruby}}== Float calculation as Tcl. <syntaxhighlight lang="ruby">include Math class Integer def permutation(k) (self-k+1 .. self).inject( :) end def combination(k) self.permutation(k) / (1 .. k).inject( :) end def big_permutation(k) exp( lgamma_plus(self) - lgamma_plus(self -k)) end def big_combination(k) exp( lgamma_plus(self) - lgamma_plus(self - k) - lgamma_plus(k)) end private def lgamma_plus(n) lgamma(n+1)[0] #lgamma is the natural log of gamma end end p 12.permutation(9) #=> 79833600 p 12.big_permutation(9) #=> 79833600.00000021 p 60.combination(53) #=> 386206920 p 145.big_permutation(133) #=> 1.6801459655817956e+243 p 900.big_combination(450) #=> 2.247471882064647e+269 p 1000.big_combination(969) #=> 7.602322407770517e+58 p 15000.big_permutation(73) #=> 6.004137561717704e+304 #That's about the maximum of Float: p 15000.big_permutation(74) #=> Infinity #Fixnum has no maximum: p 15000.permutation(74) #=> 896237613852967826239917238565433149353074416025197784301593335243699358040738127950872384197159884905490054194835376498534786047382445592358843238688903318467070575184552953997615178973027752714539513893159815472948987921587671399790410958903188816684444202526779550201576117111844818124800000000000000000000 </syntaxhighlight> Ruby's Arrays have a permutation and a combination method which result in (lazy) enumerators. These Enumerators have a "size" method, which returns the size of the enumerator, or nil if it can’t be calculated lazily. (Since Ruby 2.0) <syntaxhighlight lang="ruby">(1..60).to_a.combination(53).size #=> 386206920</syntaxhighlight> =={{header\|Rust}}== {{trans\|zig}} Almost a verbatim port... the logic is retained but I renounce anyway to use `io::stdout()`, `println!()` is enough. I wonder why f64?! <syntaxhighlight lang="rust"> fn perm(n: f64, k: f64) -> f64 { let mut result: f64 = 1.0; let mut i: f64 = 0.0; while i < k { result = n - i; i += 1.0; } result } fn comb(n: f64, k: f64) -> f64 { perm(n, k) / perm(k, k) } fn main() { const P: f64 = 12.0; const C: f64 = 60.0; let mut j: f64 = 1.0; let mut k: f64 = 10.0; while j < P { println!("P({},{}) = {}", P, j, perm(P, j).floor()); j += 1.0; } while k < C { println!("C({},{}) = {}", C, k, comb(C, k).floor()); k += 10.0; } } </syntaxhighlight> {{out}} <pre> P(12,1) = 12 P(12,2) = 132 P(12,3) = 1320 P(12,4) = 11880 P(12,5) = 95040 P(12,6) = 665280 P(12,7) = 3991680 P(12,8) = 19958400 P(12,9) = 79833600 P(12,10) = 239500800 P(12,11) = 479001600 C(60,10) = 75394027566 C(60,20) = 4191844505805495 C(60,30) = 118264581564861420 C(60,40) = 4191844505805495 C(60,50) = 75394027566 </pre> =={{header\|Scheme}}== <syntaxhighlight lang="scheme"> (define (combinations n k) (do ((i 0 (+ 1 i)) (res 1 (/ ( res (- n i)) (- k i)))) ((= i k) res))) (define (permutations n k) (do ((i 0 (+ 1 i)) (res 1 (* res (- n i)))) ((= i k) res))) (display "P(4,2) = ") (display (permutations 4 2)) (newline) (display "P(8,2) = ") (display (permutations 8 2)) (newline) (display "P(10,8) = ") (display (permutations 10 8)) (newline) (display "C(10,8) = ") (display (combinations 10 8)) (newline) (display "C(20,8) = ") (display (combinations 20 8)) (newline) (display "C(60,58) = ") (display (combinations 60 58)) (newline) (display "P(1000,10) = ") (display (permutations 1000 10)) (newline) (display "P(1000,20) = ") (display (permutations 1000 20)) (newline) (display "P(15000,2) = ") (display (permutations 15000 3)) (newline) (display "C(1000,10) = ") (display (combinations 1000 10)) (newline) (display "C(1000,999) = ") (display (combinations 1000 999)) (newline) (display "C(1000,1000) = ") (display (combinations 1000 1000)) (newline) (display "C(15000,14998) = ") (display (combinations 15000 14998)) (newline) </syntaxhighlight> {{out}} <pre> P(4,2) = 12 P(8,2) = 56 P(10,8) = 1814400 C(10,8) = 45 C(20,8) = 125970 C(60,58) = 1770 P(1000,10) = 955860613004397508326213120000 P(1000,20) = 825928413359200443640727373872992573951185652339949568000000 P(15000,2) = 3374325030000 C(1000,10) = 263409560461970212832400 C(1000,999) = 1000 C(1000,1000) = 1 C(15000,14998) = 112492500 </pre> =={{header\|Sidef}}== {{trans\|Raku}} <syntaxhighlight lang="ruby">func P(n, k) { n! / ((n-k)!) } func C(n, k) { binomial(n, k) } class Logarithm(value) { method to_s { var e = int(value/10.log) "%.8fE%+d" % (exp(value - e10.log), e) } } func lstirling(n) { n < 10 ? (lstirling(n+1) - log(n+1)) : (0.5log(2Num.pin) + nlog(n/Num.e + 1/(12Num.en))) } func P_approx(n, k) { Logarithm((lstirling(n) - lstirling(n -k))) } func C_approx(n, k) { Logarithm((lstirling(n) - lstirling(n -k) - lstirling(k))) } say "=> Exact results:" for n (1..12) { var p = n//3 say "P(#{n}, #{p}) = #{P(n, p)}" } for n (10..60 `by` 10) { var p = n//3 say "C(#{n}, #{p}) = #{C(n, p)}" } say ''; say "=> Floating point approximations:" for n ([5, 50, 500, 1000, 5000, 15000]) { var p = n//3 say "P(#{n}, #{p}) = #{P_approx(n, p)}" } for n (100..1000 `by` 100) { var p = n//3 say "C(#{n}, #{p}) = #{C_approx(n, p)}" }</syntaxhighlight> {{out}} <pre> => Exact results: P(1, 0) = 1 P(2, 0) = 1 P(3, 1) = 3 P(4, 1) = 4 P(5, 1) = 5 P(6, 2) = 30 P(7, 2) = 42 P(8, 2) = 56 P(9, 3) = 504 P(10, 3) = 720 P(11, 3) = 990 P(12, 4) = 11880 C(10, 3) = 120 C(20, 6) = 38760 C(30, 10) = 30045015 C(40, 13) = 12033222880 C(50, 16) = 4923689695575 C(60, 20) = 4191844505805495 => Floating point approximations: P(5, 1) = 5.00000000E+0 P(50, 16) = 1.03017326E+26 P(500, 166) = 3.53487492E+434 P(1000, 333) = 5.96932629E+971 P(5000, 1666) = 6.85674576E+6025 P(15000, 5000) = 9.64985399E+20469 C(100, 33) = 2.94692433E+26 C(200, 66) = 7.26975256E+53 C(300, 100) = 4.15825147E+81 C(400, 133) = 1.25794868E+109 C(500, 166) = 3.92602839E+136 C(600, 200) = 2.50601778E+164 C(700, 233) = 8.10320356E+191 C(800, 266) = 2.64562336E+219 C(900, 300) = 1.74335637E+247 C(1000, 333) = 5.77613455E+274 </pre> =={{header\|Stata}}== The '''[https://www.stata.com/help.cgi?mf_comb comb]''' function is builtin. Here is an implementation, together with perm: <syntaxhighlight lang="stata">real scalar comb1(n, k) { return(exp(lnfactorial(n)-lnfactorial(k)-lnfactorial(n-k))) } real scalar perm(n, k) { return(exp(lnfactorial(n)-lnfactorial(n-k))) }</syntaxhighlight> =={{header\|Swift}}== {{trans\|Kotlin}} Using AttaSwift's BigInt <syntaxhighlight lang="swift">import BigInt func permutations(n: Int, k: Int) -> BigInt { let l = n - k + 1 guard l <= n else { return 1 } return (l...n).reduce(BigInt(1), { $0 BigInt($1) }) } func combinations(n: Int, k: Int) -> BigInt { let fact = {() -> BigInt in guard k > 1 else { return 1 } return (2...k).map({ BigInt($0) }).reduce(1, ) }() return permutations(n: n, k: k) / fact } print("Sample of permutations from 1 to 12") for i in 1...12 { print("\(i) P \(i / 3) = \(permutations(n: i, k: i / 3))") } print("\nSample of combinations from 10 to 60") for i in stride(from: 10, through: 60, by: 10) { print("\(i) C \(i / 3) = \(combinations(n: i, k: i / 3))") } print("\nSample of permutations from 5 to 15,000") for i in [5, 50, 500, 1000, 5000, 15000] { let k = i / 3 let res = permutations(n: i, k: k).description let extra = res.count > 40 ? "... (\(res.count - 40) more digits)" : "" print("\(i) P \(k) = \(res.prefix(40))\(extra)") } print("\nSample of combinations from 100 to 1000") for i in stride(from: 100, through: 1000, by: 100) { let k = i / 3 let res = combinations(n: i, k: k).description let extra = res.count > 40 ? "... (\(res.count - 40) more digits)" : "" print("\(i) C \(k) = \(res.prefix(40))\(extra)") }</syntaxhighlight> {{out}} <pre>Sample of permutations from 1 to 12 1 P 0 = 1 2 P 0 = 1 3 P 1 = 3 4 P 1 = 4 5 P 1 = 5 6 P 2 = 30 7 P 2 = 42 8 P 2 = 56 9 P 3 = 504 10 P 3 = 720 11 P 3 = 990 12 P 4 = 11880 Sample of combinations from 10 to 60 10 C 3 = 120 20 C 6 = 38760 30 C 10 = 30045015 40 C 13 = 12033222880 50 C 16 = 4923689695575 60 C 20 = 4191844505805495 Sample of permutations from 5 to 15,000 5 P 1 = 5 50 P 16 = 103017324974226408345600000 500 P 166 = 3534874921742942787609361826601762306844... (395 more digits) 1000 P 333 = 5969326288503415089039701765900784280998... (932 more digits) 5000 P 1666 = 6856745757255674275484536940248896062234... (5986 more digits) 15000 P 5000 = 9649853988727493922014858805931295980792... (20430 more digits) Sample of combinations from 100 to 1000 100 C 33 = 294692427022540894366527900 200 C 66 = 7269752545169278341527066651192738976755... (14 more digits) 300 C 100 = 4158251463258564744783383526326405580280... (42 more digits) 400 C 133 = 1257948684182108702133348475651965004491... (70 more digits) 500 C 166 = 3926028386194422755220408345072331428197... (97 more digits) 600 C 200 = 2506017783221402805005616770513228835202... (125 more digits) 700 C 233 = 8103203563339599904740453644031138232944... (152 more digits) 800 C 266 = 2645623362683627034288829299556124255091... (180 more digits) 900 C 300 = 1743356373296446642960730765085718347630... (208 more digits) 1000 C 333 = 5776134553147651669777486323549601722339... (235 more digits)</pre> =={{header\|Tcl}}== Tcl doesn't allow the definition of new infix operators, so we define <math>P</math> and <math>C</math> as ordinary functions. There are no problems with loss of significance though: Tcl has supported arbitrary precision integer arithmetic since 8.5. {{tcllib\|math}} <~~lang~~syntaxhighlight lang="tcl"># Exact integer versions proc tcl::mathfunc::P {n k} { set t 1 Line 330 ⟶ 3,276: proc tcl::mathfunc::fC {n k} { expr {exp(lnGamma($n+1) - lnGamma($n-$k+1) - lnGamma($k+1))} }</~~lang~~syntaxhighlight> Demonstrating: <~~lang~~syntaxhighlight lang="tcl"># Using the exact integer versions puts "A sample of Permutations from 1 to 12:" for {set i 4} {$i <= 12} {incr i} { Line 357 ⟶ 3,303: set iii [expr {$i - int(sqrt($i))}] puts "$i C $ii = [expr {fC($i,$ii)}], $i C $iii = [expr {fC($i,$iii)}]" }</~~lang~~syntaxhighlight> {{out}} <pre> Line 406 ⟶ 3,352: </pre> It should be noted that for large values, it can be ''much'' faster to use the floating point version (at a cost of losing significance). In particular <code>expr C(1000,500)</code> takes approximately 1000 times longer to compute than <code>expr fC(1000,500)</code> =={{header\|VBScript}}== <syntaxhighlight lang="vb">' Combinations and permutations - vbs - 10/04/2017 dim i,j Wscript.StdOut.WriteLine "-- Long Integer - Permutations - from 1 to 12" for i=1 to 12 for j=1 to i Wscript.StdOut.Write "P(" & i & "," & j & ")=" & perm(i,j) & " " next 'j Wscript.StdOut.WriteLine "" next 'i Wscript.StdOut.WriteLine "-- Float integer - Combinations from 10 to 60" for i=10 to 60 step 10 for j=1 to i step i\5 Wscript.StdOut.Write "C(" & i & "," & j & ")=" & comb(i,j) & " " next 'j Wscript.StdOut.WriteLine "" next 'i Wscript.StdOut.WriteLine "-- Float integer - Permutations from 5000 to 15000" for i=5000 to 15000 step 5000 for j=10 to 70 step 20 Wscript.StdOut.Write "C(" & i & "," & j & ")=" & perm(i,j) & " " next 'j Wscript.StdOut.WriteLine "" next 'i Wscript.StdOut.WriteLine "-- Float integer - Combinations from 200 to 1000" for i=200 to 1000 step 200 for j=20 to 100 step 20 Wscript.StdOut.Write "P(" & i & "," & j & ")=" & comb(i,j) & " " next 'j Wscript.StdOut.WriteLine "" next 'i function perm(x,y) dim i,z z=1 for i=x-y+1 to x z=zi next 'i perm=z end function 'perm function fact(x) dim i,z z=1 for i=2 to x z=zi next 'i fact=z end function 'fact function comb(byval x,byval y) if y>x then comb=0 elseif x=y then comb=1 else if x-y<y then y=x-y comb=perm(x,y)/fact(y) end if end function 'comb</syntaxhighlight> {{out}} <pre style="height:40ex"> -- Long Integer - Permutations - from 1 to 12 P(1,1)=1 P(2,1)=2 P(2,2)=2 P(3,1)=3 P(3,2)=6 P(3,3)=6 P(4,1)=4 P(4,2)=12 P(4,3)=24 P(4,4)=24 P(5,1)=5 P(5,2)=20 P(5,3)=60 P(5,4)=120 P(5,5)=120 P(6,1)=6 P(6,2)=30 P(6,3)=120 P(6,4)=360 P(6,5)=720 P(6,6)=720 P(7,1)=7 P(7,2)=42 P(7,3)=210 P(7,4)=840 P(7,5)=2520 P(7,6)=5040 P(7,7)=5040 P(8,1)=8 P(8,2)=56 P(8,3)=336 P(8,4)=1680 P(8,5)=6720 P(8,6)=20160 P(8,7)=40320 P(8,8)=40320 P(9,1)=9 P(9,2)=72 P(9,3)=504 P(9,4)=3024 P(9,5)=15120 P(9,6)=60480 P(9,7)=181440 P(9,8)=362880 P(9,9)=362880 P(10,1)=10 P(10,2)=90 P(10,3)=720 P(10,4)=5040 P(10,5)=30240 P(10,6)=151200 P(10,7)=604800 P(10,8)=1814400 P(10,9)=3628800 P(10,10)=3628800 P(11,1)=11 P(11,2)=110 P(11,3)=990 P(11,4)=7920 P(11,5)=55440 P(11,6)=332640 P(11,7)=1663200 P(11,8)=6652800 P(11,9)=19958400 P(11,10)=39916800 P(11,11)=39916800 P(12,1)=12 P(12,2)=132 P(12,3)=1320 P(12,4)=11880 P(12,5)=95040 P(12,6)=665280 P(12,7)=3991680 P(12,8)=19958400 P(12,9)=79833600 P(12,10)=239500800 P(12,11)=479001600 P(12,12)=479001600 -- Float integer - Combinations from 10 to 60 C(10,1)=10 C(10,3)=120 C(10,5)=252 C(10,7)=120 C(10,9)=10 C(20,1)=20 C(20,5)=15504 C(20,9)=167960 C(20,13)=77520 C(20,17)=1140 C(30,1)=30 C(30,7)=2035800 C(30,13)=119759850 C(30,19)=54627300 C(30,25)=142506 C(40,1)=40 C(40,9)=273438880 C(40,17)=88732378800 C(40,25)=40225345056 C(40,33)=18643560 C(50,1)=50 C(50,11)=37353738800 C(50,21)=67327446062800 C(50,31)=30405943383200 C(50,41)=2505433700 C(60,1)=60 C(60,13)=5166863427600 C(60,25)=5,19154379743283E+16 C(60,37)=2,33853324208686E+16 C(60,49)=342700125300 -- Float integer - Permutations from 5000 to 15000 C(5000,10)=9,67807348145655E+36 C(5000,30)=8,53575581200676E+110 C(5000,50)=6,94616656703754E+184 C(5000,70)=5,21383580146194E+258 C(10000,10)=9,95508690556325E+39 C(10000,30)=9,57391540294832E+119 C(10000,50)=8,84526658067387E+199 C(10000,70)=7,850079552152E+279 C(15000,10)=5,74922667554068E+41 C(15000,30)=1,86266591363916E+125 C(15000,50)=5,87565776023335E+208 C(15000,70)=1,80450662858719E+292 -- Float integer - Combinations from 200 to 1000 P(200,20)=1,61358778796735E+27 P(200,40)=2,05015799519859E+42 P(200,60)=7,04050484926892E+51 P(200,80)=1,64727865245176E+57 P(200,100)=9,05485146561033E+58 P(400,20)=2,7883609836709E+33 P(400,40)=1,9703374084393E+55 P(400,60)=1,50867447857277E+72 P(400,80)=4,22814216193593E+85 P(400,100)=2,24185479155434E+96 P(600,20)=1,09108668819553E+37 P(600,40)=4,33518929550349E+62 P(600,60)=2,77426667704894E+83 P(600,80)=1,00412999166192E+101 P(600,100)=1,11141121906619E+116 P(800,20)=3,72976760205571E+39 P(800,40)=6,04464684067502E+67 P(800,60)=1,90370080982158E+91 P(800,80)=4,14170924105943E+111 P(800,100)=3,4111376846871E+129 P(1000,20)=3,39482811302458E+41 P(1000,40)=5,55974423571664E+71 P(1000,60)=1,97427486218598E+97 P(1000,80)=5,43269728730706E+119 P(1000,100)=6,38505119263051E+139 </pre> =={{header\|Visual Basic .NET}}== {{works with\|Visual Basic .NET\|2013}} <syntaxhighlight lang="vbnet">' Combinations and permutations - 10/04/2017 Imports System.Numerics 'BigInteger Module CombPermRc Sub Main() Dim i, j As Long For i = 1 To 12 For j = 1 To i Console.Write("P(" & i & "," & j & ")=" & PermBig(i, j).ToString & " ") Next j Console.WriteLine("") Next i Console.WriteLine("--") For i = 10 To 60 Step 10 For j = 1 To i Step i \ 5 Console.Write("C(" & i & "," & j & ")=" & CombBig(i, j).ToString & " ") Next j Console.WriteLine("") Next i Console.WriteLine("--") For i = 5000 To 15000 Step 5000 For j = 4000 To 5000 Step 1000 Console.Write("P(" & i & "," & j & ")=" & PermBig(i, j).ToString("E") & " ") Next j Console.WriteLine("") Next i Console.WriteLine("--") For i = 5000 To 15000 Step 5000 For j = 4000 To 5000 Step 1000 Console.Write("C(" & i & "," & j & ")=" & CombBig(i, j).ToString("E") & " ") Next j Console.WriteLine("") Next i Console.WriteLine("--") i = 5000 : j = 4000 Console.WriteLine("C(" & i & "," & j & ")=" & CombBig(i, j).ToString) End Sub 'Main Function PermBig(x As Long, y As Long) As BigInteger Dim i As Long, z As BigInteger z = 1 For i = x - y + 1 To x z = z i Next i Return (z) End Function 'PermBig Function FactBig(x As Long) As BigInteger Dim i As Long, z As BigInteger z = 1 For i = 2 To x z = z * i Next i Return (z) End Function 'FactBig Function CombBig(ByVal x As Long, ByVal y As Long) As BigInteger If y > x Then Return (0) ElseIf x = y Then Return (1) Else If x - y < y Then y = x - y Return (PermBig(x, y) / FactBig(y)) End If End Function 'CombBig End Module</syntaxhighlight> {{out}} <pre style="height:40ex"> P(1,1)=1 P(2,1)=2 P(2,2)=2 P(3,1)=3 P(3,2)=6 P(3,3)=6 P(4,1)=4 P(4,2)=12 P(4,3)=24 P(4,4)=24 P(5,1)=5 P(5,2)=20 P(5,3)=60 P(5,4)=120 P(5,5)=120 P(6,1)=6 P(6,2)=30 P(6,3)=120 P(6,4)=360 P(6,5)=720 P(6,6)=720 P(7,1)=7 P(7,2)=42 P(7,3)=210 P(7,4)=840 P(7,5)=2520 P(7,6)=5040 P(7,7)=5040 P(8,1)=8 P(8,2)=56 P(8,3)=336 P(8,4)=1680 P(8,5)=6720 P(8,6)=20160 P(8,7)=40320 P(8,8)=40320 P(9,1)=9 P(9,2)=72 P(9,3)=504 P(9,4)=3024 P(9,5)=15120 P(9,6)=60480 P(9,7)=181440 P(9,8)=362880 P(9,9)=362880 P(10,1)=10 P(10,2)=90 P(10,3)=720 P(10,4)=5040 P(10,5)=30240 P(10,6)=151200 P(10,7)=604800 P(10,8)=1814400 P(10,9)=3628800 P(10,10)=3628800 P(11,1)=11 P(11,2)=110 P(11,3)=990 P(11,4)=7920 P(11,5)=55440 P(11,6)=332640 P(11,7)=1663200 P(11,8)=6652800 P(11,9)=19958400 P(11,10)=39916800 P(11,11)=39916800 P(12,1)=12 P(12,2)=132 P(12,3)=1320 P(12,4)=11880 P(12,5)=95040 P(12,6)=665280 P(12,7)=3991680 P(12,8)=19958400 P(12,9)=79833600 P(12,10)=239500800 P(12,11)=479001600 P(12,12)=479001600 -- C(10,1)=10 C(10,3)=120 C(10,5)=252 C(10,7)=120 C(10,9)=10 C(20,1)=20 C(20,5)=15504 C(20,9)=167960 C(20,13)=77520 C(20,17)=1140 C(30,1)=30 C(30,7)=2035800 C(30,13)=119759850 C(30,19)=54627300 C(30,25)=142506 C(40,1)=40 C(40,9)=273438880 C(40,17)=88732378800 C(40,25)=40225345056 C(40,33)=18643560 C(50,1)=50 C(50,11)=37353738800 C(50,21)=67327446062800 C(50,31)=30405943383200 C(50,41)=2505433700 C(60,1)=60 C(60,13)=5166863427600 C(60,25)=51915437974328292 C(60,37)=23385332420868600 C(60,49)=342700125300 -- P(5000,4000)=1,050873E+13758 P(5000,5000)=4,228578E+16325 P(10000,4000)=1,060455E+15594 P(10000,5000)=6,731009E+19333 P(15000,4000)=8,685001E+16448 P(15000,5000)=9,649854E+20469 -- C(5000,4000)=5,746236E+1084 C(5000,5000)=1,000000E+000 C(10000,4000)=5,798630E+2920 C(10000,5000)=1,591790E+3008 C(15000,4000)=4,749011E+3775 C(15000,5000)=2,282057E+4144 -- C(5000,4000)=57462357505803375604893834658665168251899919793850512934468881710397678593302188064618445132583370701755893065787216750992391223467601994741594656878559929037277303674963658032197224327768110236651567704673226756781828332650887849150208195780031161286578505113618731045004523840401144118298192191997565735245181433457469532981432785237769191864102953974244072964471109551273603780184330987071947790993108191904370472373403157802158903129815170101708451875442019845175637901995588390614304812103202403626211504997668649346891167495657556154392183988627948442807346603688457854135114491955258804187129028256547543888109987151649038111791932035229202856007767332717845596528598314477979861265222941138323298702349967224867703420888363395662988291273283611081068577160905840445308086112429900453394212790633910614322699210850302387512579976209123523546689147207974269396548873838155355768985614160932799226261509866933143702889005270480654844312094564956178277998090168826124850606847021667494019587077659107276117413835912767949017954979839258481340540145909025953956582025656306426226560 </pre> =={{header\|Wren}}== {{trans\|Kotlin}} {{libheader\|Wren-big}} {{libheader\|Wren-fmt}} {{libheader\|Wren-iterate}} <syntaxhighlight lang="wren">import "./big" for BigInt import "./fmt" for Fmt import "./iterate" for Stepped var perm = Fn.new { \|n, k\| if (n <= 0 \|\| k < 0) Fiber.abort("Invalid argument(s).") if (k == 0) return BigInt.one return (n-k+1..n).reduce(BigInt.one) { \|acc, i\| acc * BigInt.new(i) } } var comb = Fn.new { \|n, k\| if (n <= 0 \|\| k < 0) Fiber.abort("Invalid argument(s).") var fact = BigInt.one if (k > 1) fact = (2..k).reduce(BigInt.one) { \|acc, i\| acc * BigInt.new(i) } return perm.call(n, k) / fact } System.print("A sample of permutations from 1 to 12:") for (n in 1..12) Fmt.print("$2d P $-2d = $i", n, (n/3).floor, perm.call(n, (n/3).floor)) System.print("\nA sample of combinations from 10 to 60:") for (n in Stepped.new(10..60, 10)) { Fmt.print("$2d C $-2d = $i", n, (n/3).floor, comb.call(n, (n/3).floor)) } System.print("\nA sample of permutations from 5 to 15000:") var na = [5, 50, 500, 1000, 5000, 15000] for (n in na) { var k = (n/3).floor var s = perm.call(n, k).toString var l = s.count var e = (l <= 40) ? "" : "... (%(l - 40) more digits)" Fmt.print("$5d P $-4d = $s$s", n, k, s.take(40).join(), e) } System.print("\nA sample of combinations from 100 to 1000:") for (n in Stepped.new(100..1000, 100)) { var k = (n/3).floor var s = comb.call(n, k).toString var l = s.count var e = (l <= 40) ? "" : "... (%(l - 40) more digits)" Fmt.print("$4d C $-3d = $s$s", n, k, s.take(40).join(), e) }</syntaxhighlight> {{out}} <pre> A sample of permutations from 1 to 12: 1 P 0 = 1 2 P 0 = 1 3 P 1 = 3 4 P 1 = 4 5 P 1 = 5 6 P 2 = 30 7 P 2 = 42 8 P 2 = 56 9 P 3 = 504 10 P 3 = 720 11 P 3 = 990 12 P 4 = 11880 A sample of combinations from 10 to 60: 10 C 3 = 120 20 C 6 = 38760 30 C 10 = 30045015 40 C 13 = 12033222880 50 C 16 = 4923689695575 60 C 20 = 4191844505805495 A sample of permutations from 5 to 15000: 5 P 1 = 5 50 P 16 = 103017324974226408345600000 500 P 166 = 3534874921742942787609361826601762306844... (395 more digits) 1000 P 333 = 5969326288503415089039701765900784280998... (932 more digits) 5000 P 1666 = 6856745757255674275484536940248896062234... (5986 more digits) 15000 P 5000 = 9649853988727493922014858805931295980792... (20430 more digits) A sample of combinations from 100 to 1000: 100 C 33 = 294692427022540894366527900 200 C 66 = 7269752545169278341527066651192738976755... (14 more digits) 300 C 100 = 4158251463258564744783383526326405580280... (42 more digits) 400 C 133 = 1257948684182108702133348475651965004491... (70 more digits) 500 C 166 = 3926028386194422755220408345072331428197... (97 more digits) 600 C 200 = 2506017783221402805005616770513228835202... (125 more digits) 700 C 233 = 8103203563339599904740453644031138232944... (152 more digits) 800 C 266 = 2645623362683627034288829299556124255091... (180 more digits) 900 C 300 = 1743356373296446642960730765085718347630... (208 more digits) 1000 C 333 = 5776134553147651669777486323549601722339... (235 more digits) </pre> =={{header\|zig}}== <syntaxhighlight lang="zig> const std = @import("std"); const num = f64; pub fn perm(n: num, k: num) num { var result: num = 1; var i: num = 0; while (i < k) : (i += 1) { result *= n - i; } return result; } pub fn comb(n: num, k: num) num { return perm(n, k) / perm(k, k); } pub fn main() !void { var stdout = std.io.getStdOut().writer(); const p: num = 12; const c: num = 60; var j: num = 1; var k: num = 10; while (j < p) : (j += 1) { try stdout.print("P({d},{d}) = {d}\n", .{ p, j, @floor(perm(p, j)) }); } while (k < c) : (k += 10) { try stdout.print("C({d},{d}) = {d}\n", .{ c, k, @floor(comb(c, k)) }); } }</syntaxhighlight> {{out}} <pre> P(12,1) = 12 P(12,2) = 132 P(12,3) = 1320 P(12,4) = 11880 P(12,5) = 95040 P(12,6) = 665280 P(12,7) = 3991680 P(12,8) = 19958400 P(12,9) = 79833600 P(12,10) = 239500800 P(12,11) = 479001600 C(60,10) = 75394027566 C(60,20) = 4191844505805495 C(60,30) = 118264581564861420 C(60,40) = 4191844505805495 C(60,50) = 75394027566 </pre>