Combinations and permutations: Difference between revisions
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Compiled with <code>g++-10 |
Compiled with <code>g++-10 -Wall -std=c++03</code>, linked with <code>-lgmp</code> |
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<lang cpp>#include <boost/multiprecision/gmp.hpp> |
<lang cpp>#include <boost/multiprecision/gmp.hpp> |
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#include <iostream> |
#include <iostream> |
Revision as of 09:59, 19 September 2020
You are encouraged to solve this task according to the task description, using any language you may know.
This page uses content from Wikipedia. The original article was at Combination. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
This page uses content from Wikipedia. The original article was at Permutation. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
- Task
Implement the combination (nCk) and permutation (nPk) operators in the target language:
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.
This 'floating point' code could be implemented using an approximation, e.g., by calling the Gamma function.
- Related task
The number of samples of size k from n objects.
With combinations and permutations generation tasks.
Order Unimportant Order Important Without replacement Task: Combinations Task: Permutations With replacement Task: Combinations with repetitions Task: Permutations with repetitions
ALGOL 68
File: prelude_combinations_and_permutations.a68<lang algol68># -*- coding: utf-8 -*- #
COMMENT REQUIRED by "prelude_combinations_and_permutations.a68" CO
MODE CPINT = #LONG# ~; MODE CPOUT = #LONG# ~; # the answer, can be REAL # MODE CPREAL = ~; # the answer, can be REAL # PROC cp fix value error = (#REF# CPARGS args)BOOL: ~;
- PROVIDES:#
- OP C = (CP~,CP~)CP~: ~ #
- OP P = (CP~,CP~)CP~: ~ #
END COMMENT
MODE CPARGS = STRUCT(CHAR name, #REF# CPINT n,k);
PRIO C = 8, P = 8; # should be 7.5, a priority between *,/ and **,SHL,SHR etc #
- I suspect there is a more reliable way of doing this using the Gamma Function approx #
OP P = (CPINT n, r)CPOUT: (
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)! #
FOR i FROM n-r+1 TO n DO out *:= i OD; out
);
OP P = (CPREAL n, r)CPREAL: # 'ln gamma' requires GSL library #
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)! #
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)! #
CPREAL i := n-r+1; WHILE i <= n DO out*:= i;
- a crude check for underflow #
IF i = i + 1 THEN IF NOT cp fix value error(CPARGS("P",ENTIER n,ENTIER r)) THEN stop FI FI; i+:=1 OD; out
); END COMMENT
- basically C(n,r) = nCk = nPk/r! = n!/(n-r)!/r! #
OP C = (CPINT n, r)CPOUT: (
IF n < r ORF r < 0 THEN IF NOT cp fix value error(("C",n,r)) THEN stop FI FI; CPINT largest = ( r > n - r | r | n - r ); CPINT smallest = n - largest; CPOUT out := 1; INT smaller fact := 2; FOR larger fact FROM largest+1 TO n DO
- try and prevent overflow, p.s. there must be a smarter way to do this #
- Problems: loop stalls when 'smaller fact' is a largeish co prime #
out *:= larger fact; WHILE smaller fact <= smallest ANDF out MOD smaller fact = 0 DO out OVERAB smaller fact; smaller fact +:= 1 OD OD; out # EXIT with: n P r OVER r P r #
);
OP C = (CPREAL n, CPREAL r)CPREAL: # 'ln gamma' requires GSL library #
exp(ln gamma(n+1)-ln gamma(n-r+1)-ln gamma(r+1));
- basically C(n,r) = nCk = nPk/r! = n!/(n-r)!/r! #
COMMENT # alternate slower version # OP C = (CPREAL n, REAL r)CPREAL: (
IF n < r ORF r < 0 THEN IF NOT cp fix value error(("C",ENTIER n,ENTIER r)) THEN stop FI FI; CPREAL largest = ( r > n - r | r | n - r ); CPREAL smallest = n - largest; CPREAL out := 1; REAL smaller fact := 2; REAL larger fact := largest+1; WHILE larger fact <= n DO # todo: check underflow here #
- try and prevent overflow, p.s. there must be a smarter way to do this #
out *:= larger fact; WHILE smaller fact <= smallest ANDF out > smaller fact DO out /:= smaller fact; smaller fact +:= 1 OD; larger fact +:= 1 OD; out # EXIT with: n P r OVER r P r #
); END COMMENT
SKIP</lang>File: test_combinations_and_permutations.a68<lang algol68>#!/usr/bin/a68g --script #
- -*- coding: utf-8 -*- #
CO REQUIRED by "prelude_combinations_and_permutations.a68" CO
MODE CPINT = #LONG# INT; MODE CPOUT = #LONG# INT; # the answer, can be REAL # MODE CPREAL = REAL; # the answer, can be REAL # PROC cp fix value error = (#REF# CPARGS args)BOOL: ( putf(stand error, ($"Value error: "g(0)gg(0)"arg out of range"l$, n OF args, name OF args, k OF args)); FALSE # unfixable # );
- PROVIDES:#
- OP C = (CP~,CP~)CP~: ~ #
- OP P = (CP~,CP~)CP~: ~ #
PR READ "prelude_combinations_and_permutations.a68" PR;
printf($"A sample of Permutations from 1 to 12:"l$); FOR i FROM 4 BY 1 TO 12 DO
INT first = i - 2, second = i - ENTIER sqrt(i); printf(($g(0)" P "g(0)" = "g(0)$, i, first, i P first, $", "$)); printf(($g(0)" P "g(0)" = "g(0)$, i, second, i P second, $l$))
OD;
printf($l"A sample of Combinations from 10 to 60:"l$); FOR i FROM 10 BY 10 TO 60 DO
INT first = i - 2, second = i - ENTIER sqrt(i); printf(($"("g(0)" C "g(0)") = "g(0)$, i, first, i C first, $", "$)); printf(($"("g(0)" C "g(0)") = "g(0)$, i, second, i C second, $l$))
OD;
printf($l"A sample of Permutations from 5 to 15000:"l$); FOR i FROM 5 BY 10 TO 150 DO
REAL r = i, first = r - 2, second = r - ENTIER sqrt(r); printf(($g(0)" P "g(0)" = "g(-real width,real width-5,-1)$, r, first, r P first, $", "$)); printf(($g(0)" P "g(0)" = "g(-real width,real width-5,-1)$, r, second, r P second, $l$))
OD;
printf($l"A sample of Combinations from 10 to 190:"l$); FOR i FROM 100 BY 100 TO 1000 DO
REAL r = i, first = r - 2, second = r - ENTIER sqrt(r); 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>Output:
A sample of Permutations from 1 to 12: 4 P 2 = 12, 4 P 2 = 12 5 P 3 = 60, 5 P 3 = 60 6 P 4 = 360, 6 P 4 = 360 7 P 5 = 2520, 7 P 5 = 2520 8 P 6 = 20160, 8 P 6 = 20160 9 P 7 = 181440, 9 P 6 = 60480 10 P 8 = 1814400, 10 P 7 = 604800 11 P 9 = 19958400, 11 P 8 = 6652800 12 P 10 = 239500800, 12 P 9 = 79833600 A sample of Combinations from 10 to 60: (10 C 8) = 45, (10 C 7) = 120 (20 C 18) = 190, (20 C 16) = 4845 (30 C 28) = 435, (30 C 25) = 142506 (40 C 38) = 780, (40 C 34) = 3838380 (50 C 48) = 1225, (50 C 43) = 99884400 (60 C 58) = 1770, (60 C 53) = 386206920 A sample of Permutations from 5 to 15000: 5 P 3 = 6.0000000000e1, 5 P 3 = 6.0000000000e1 15 P 13 = 6.538371840e11, 15 P 12 = 2.179457280e11 25 P 23 = 7.755605022e24, 25 P 20 = 1.292600837e23 35 P 33 = 5.166573983e39, 35 P 30 = 8.610956639e37 45 P 43 = 5.981111043e55, 45 P 39 = 1.661419734e53 55 P 53 = 6.348201677e72, 55 P 48 = 2.519127650e69 65 P 63 = 4.123825296e90, 65 P 57 = 2.045548262e86 75 P 73 = 1.24045704e109, 75 P 67 = 6.15306072e104 85 P 83 = 1.40855206e128, 85 P 76 = 7.76318374e122 95 P 93 = 5.16498924e147, 95 P 86 = 2.84666515e142 105 P 103 = 5.40698379e167, 105 P 95 = 2.98003957e161 115 P 113 = 1.46254685e188, 115 P 105 = 8.06077407e181 125 P 123 = 9.41338588e208, 125 P 114 = 4.71650327e201 135 P 133 = 1.34523635e230, 135 P 124 = 6.74020139e222 145 P 143 = 4.02396303e251, 145 P 133 = 1.68014597e243 A sample of Combinations from 10 to 190: (100 C 98) = 4950.0, (100 C 90) = 17310309456438.8 (200 C 198) = 19900.0, (200 C 186) = 1179791641436960000000.0 (300 C 298) = 44850.0, (300 C 283) = 2287708142022840000000000000.0 (400 C 398) = 79800.0, (400 C 380) = 2788360983670300000000000000000000.0 (500 C 498) = 124750.0, (500 C 478) = 132736424690773000000000000000000000000.0 (600 C 598) = 179700.0, (600 C 576) = 4791686682467800000000000000000000000000000.0 (700 C 698) = 244650.0, (700 C 674) = 145478651313640000000000000000000000000000000000.0 (800 C 798) = 319600.0, (800 C 772) = 3933526871034430000000000000000000000000000000000000.0 (900 C 898) = 404550.0, (900 C 870) = 98033481673646900000000000000000000000000000000000000000.0 (1000 C 998) = 499500.0, (1000 C 969) = 76023224077705100000000000000000000000000000000000000000000.0
Bracmat
Bracmat cannot handle floating point numbers. Instead, this solution shows the first 50 digits and a count of the digits that are not shown. <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 )
& ( P
= n k result . !arg:(?n,?k) & !n+-1*!k:?k & 1:?result & whl ' ( !n:>!k & !n*!result:?result & !n+-1:?n ) & !result )
& 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$(!i C !k "=" C$(!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$(!i " P " !k " = " displayBig$(P$(!i,!k)))) )
& 0:?i & whl
' ( 100+!i:~>1000:?i & div$(!i.3):?k & out$(str$(!i " C " !k " = " displayBig$(C$(!i,!k)))) )
);</lang> Output:
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 C 3 = 120 20 C 6 = 38760 30 C 10 = 30045015 40 C 13 = 12033222880 50 C 16 = 4923689695575 60 C 20 = 4191844505805495 5 P 1 = 5 50 P 16 = 103017324974226408345600000 500 P 166 = 35348749217429427876093618266017623068440028791060... (385 more digits) 1000 P 333 = 59693262885034150890397017659007842809981894765670... (922 more digits) 5000 P 1666 = 68567457572556742754845369402488960622341567102448... (5976 more digits) 15000 P 5000 = 96498539887274939220148588059312959807922816886808... (20420 more digits) 100 C 33 = 294692427022540894366527900 200 C 66 = 72697525451692783415270666511927389767550269141935... (4 more digits) 300 C 100 = 41582514632585647447833835263264055802804660057436... (32 more digits) 400 C 133 = 12579486841821087021333484756519650044917494358375... (60 more digits) 500 C 166 = 39260283861944227552204083450723314281973490135301... (87 more digits) 600 C 200 = 25060177832214028050056167705132288352025510250879... (115 more digits) 700 C 233 = 81032035633395999047404536440311382329449203119421... (142 more digits) 800 C 266 = 26456233626836270342888292995561242550915240450150... (170 more digits) 900 C 300 = 17433563732964466429607307650857183476303419689548... (198 more digits) 1000 C 333 = 57761345531476516697774863235496017223394580195002... (225 more digits)
C
Using big integers. GMP in fact has a factorial function which is quite possibly more efficient, though using it would make code longer. <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; }</lang>
C++
Compiled with g++-10 -Wall -std=c++03
, linked with -lgmp
<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;
}</lang>
- Output:
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
Common Lisp
<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))))) </lang>
Crystal
<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
</lang>
- Output:
79833600 79833600.00000021 386206920 1.6801459655817956e+243 2.247471882064647e+269 7.602322407770517e+58 6.004137561717704e+304 Infinity 896237613852967826239917238565433149353074416025197784301593335243699358040738127950872384197159884905490054194835376498534786047382445592358843238688903318467070575184552953997615178973027752714539513893159815472948987921587671399790410958903188816684444202526779550201576117111844818124800000000000000000000
D
<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));
}</lang>
- Output:
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
EchoLisp
<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 </lang>
Elixir
<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</lang>
- Output:
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)
Erlang
<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.
</lang>
Output:
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)
Factor
As with Racket, these operations are built in and Factor has unlimited integers. <lang factor>USING: math.combinatorics prettyprint ;
1000 10 nCk . ! 263409560461970212832400 1000 10 nPk . ! 955860613004397508326213120000</lang>
Go
Go has arbitrary-length maths in the standard math/big library; no need for floating-point approximations at this level. <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 } </lang> Output:
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
Haskell
The Haskell Integer type supports arbitrary precision so floating point approximation is not needed. <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] ]
</lang>
- Output:
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)
Icon and 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.
<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</lang>
Output:
->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 ->
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:
<lang J>C=: ! P=: (%&!&x:~ * <:)"0</lang>
! is a primitive, but we will give it a name (C
) for this task.
Example use (P is permutations, C is combinations):
<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+10*i.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</lang>
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”. <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; }
} </lang>
- Output:
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
jq
Currently, jq approximates large integers by IEEE 754 64-bit floats, and only supports tgamma (true gamma). Thus beyond about 1e308 all accuracy is lost. <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;</lang> Examples:
12 | permutation(9) #=> 79833600
12 | big_permutation(9) #=> 79833599.99999964
60 | combination(53) #=> 386206920
60 | big_combination(53) #=> 386206920.0000046
145 | big_permutation(133) #=> 1.6801459655817e+243
170 | big_permutation(133) #=> 5.272846415658284e+263
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 FloatingPoint versions of these operators for large numbers would be to use arbitrary precision integers, BigInt.
Functions <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) </lang>
Main <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 </lang>
- Output:
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
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. <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) }
}</lang>
- Output:
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)
M2000 Interpreter
<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 </lang>
- Output:
-- 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
Maple
<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; </lang>
Mathematica / Wolfram Language
<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</lang>
- Output:
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 2.011936300385469*10^2567 5000 P 4998 2.114288963302772*10^16325 10000 P 9998 1.423129840458527*10^35659 15000 P 14998 1.373299516742584*10^56129 100 C 50 1.00891*10^29 200 C 100 9.05485*10^58 300 C 150 9.37597*10^88 400 C 200 1.02953*10^119 500 C 250 1.16744*10^149 600 C 300 1.35108*10^179 700 C 350 1.58574*10^209 800 C 400 1.88042*10^239 900 C 450 2.24747*10^269 1000 C 500 2.70288*10^299
Note that Mathematica can easily handle very big numbers with exact integer arithmetic: <lang Mathematica> Permutation[200000, 100000] </lang>
- Output:
The output is 516777 digits longs:
50287180689616781338617355322585606........0321815299686400000000000000000000......(lots of zeroes)
МК-61/52
<lang>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</lang>
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.
Nim
<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)</lang>
PARI/GP
<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);</lang>
- Output:
?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
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.
<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; } </lang>
Since the output is almost the same as Raku's, and this is only a Draft RosettaCode task, I'm not going to bother including the output of the program.
Phix
Translation of Raku/Sidef, same results
Update: there are now builtin routines k_perm(n,k) and choose(n,k), slightly more efficient equivalents of P() and C() respectively.
<lang Phix>function P(integer n,k)
return factorial(n)/factorial(n-k)
end function
function C(integer n,k)
return P(n,k)/factorial(k)
end function
function lstirling(atom n)
if n<10 then return lstirling(n+1)-log(n+1) end if return 0.5*log(2*PI*n) + n*log(n/E + 1/(12*E*n))
end function
function P_approx(integer n, k)
return lstirling(n)-lstirling(n-k)
end function
function C_approx(integer n, k)
return lstirling(n)-lstirling(n-k)-lstirling(k)
end function
function to_s(atom v) integer e = floor(v/log(10))
return sprintf("%.9ge%d",{power(E,v-e*log(10)),e})
end function</lang> Test code <lang Phix>printf(1,"=> Exact results:\n") for n=1 to 12 do
integer p = floor(n/3) printf(1,"P(%d,%d) = %d\n",{n,p,P(n,p)})
end for
for n=10 to 60 by 10 do
integer p = floor(n/3) printf(1,"C(%d,%d) = %d\n",{n,p,C(n,p)})
end for
printf(1,"=> Floating point approximations:\n") constant tests = {5, 50, 500, 1000, 5000, 15000} for i=1 to length(tests) do
integer n=tests[i], p = floor(n/3) printf(1,"P(%d,%d) = %s\n",{n,p,to_s(P_approx(n,p))})
end for
for n=100 to 1000 by 100 do
integer p = floor(n/3) printf(1,"C(%d,%d) = %s\n",{n,p,to_s(C_approx(n,p))})
end for</lang>
Python
<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))
</lang>
- Output:
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
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:
<lang Racket>
- lang racket
(require math) (define C binomial) (define P permutations)
(C 1000 10) ; -> 263409560461970212832400 (P 1000 10) ; -> 955860613004397508326213120000 </lang>
(I'll spare this page from yet another big listing of samples...)
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 :
Notice that Raku can process arbitrary long integers, though. So it's not clear whether using floating points is useful in this case.
<lang perl6>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) !! .5*log(2*pi*n)+ n*log(n/e+1/(12*e*n))
}
role Logarithm {
method gist {
my $e = (self/10.log).Int; sprintf "%.8fE%+d", exp(self - $e*10.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);
}</lang>
- Output:
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
REXX
The hard part of this REXX program was coding the DO loops for the various ranges. <lang rexx>/*REXX program compute and displays a sampling of combinations and permutations. */ numeric digits 100 /*use 100 decimal digits of precision. */
do j=1 for 12; _= /*show all permutations from 1 ──► 12.*/ do k=1 for j /*step through all J permutations. */ _=_ 'P('j","k')='perm(j,k)" " /*add an extra blank between numbers. */ end /*k*/ say strip(_) /*show the permutations horizontally. */ end /*j*/
say /*display a blank line for readability.*/
do j=10 to 60 by 10; _= /*show some combinations 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 numbers. */ end /*k*/ say strip(_) /*show the combinations horizontally. */ end /*j*/
say /*display a blank line for readability.*/ numeric digits 20 /*force floating point for big numbers.*/
do j=5 to 15000 by 1000; _= /*show a few permutations, big numbers.*/ do k=1 to j for 5 by j%10 /*step through some J permutations. */ _=_ 'P('j","k')='perm(j,k)" " /*add an extra blank between numbers. */ end /*k*/ say strip(_) /*show the permutations horizontally. */ end /*j*/
say /*display a blank line for readability.*/
do j=100 to 1000 by 100; _= /*show a few combinations, big numbers.*/ do k= 1 to j by j%5 /*step through some combinations. */ _=_ 'C('j","k')='comb(j,k)" " /*add an extra blank between numbers. */ end /*k*/ say strip(_) /*show the combinations horizontally. */ end /*j*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ perm: procedure; parse arg x,y; call .combPerm; return _ .combPerm: _=1; do j=x-y+1 to x; _=_*j; end; return _ !: procedure; parse arg x; !=1; do j=2 to x; !=!*j; end; return ! /*──────────────────────────────────────────────────────────────────────────────────────*/ comb: procedure; parse arg x,y /*arguments: X things, Y at-a-time.*/
if y >x then return 0 /*oops-say, an error, too big a chunk.*/ if x =y then return 1 /*X things are the same as chunk size.*/ if x-y <y then y=x - y /*switch things around for speed. */ call .combPerm /*call subroutine to do heavy lifting. */ return _ / !(y) /*just perform one last division. */</lang>
output
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(5,1)=5 P(5,1)=5 P(5,1)=5 P(5,1)=5 P(5,1)=5 P(1005,1)=1005 P(1005,101)=9.1176524923776877363E+300 P(1005,201)=1.2772738260896333926E+594 P(1005,301)=6.9244021230613662196E+881 P(1005,401)=2.4492580742838357278E+1163 P(2005,1)=2005 P(2005,201)=1.6533543480914610058E+659 P(2005,401)=3.0753126526205309249E+1305 P(2005,601)=7.9852540678709597130E+1940 P(2005,801)=8.0516979630356802995E+2563 P(3005,1)=3005 P(3005,301)=1.1935689764209015622E+1040 P(3005,601)=1.5619600532077469150E+2062 P(3005,901)=1.0291767405881430479E+3068 P(3005,1201)=1.5669988662999668720E+4055 P(4005,1)=4005 P(4005,401)=4.3808609526948101266E+1435 P(4005,801)=2.3060742016678396933E+2848 P(4005,1201)=2.2044072986703009755E+4239 P(4005,1601)=2.8973897505902543204E+5605 P(5005,1)=5005 P(5005,501)=1.4180262672357809801E+1842 P(5005,1001)=2.8239356430641573722E+3656 P(5005,1501)=3.6832518654277810594E+5443 P(5005,2001)=3.9303728189857603162E+7199 P(6005,1)=6005 P(6005,601)=2.4482219222979658097E+2257 P(6005,1201)=1.0247320583108167487E+4482 P(6005,1801)=1.0131515875595211375E+6674 P(6005,2401)=4.8762853043004294329E+8828 P(7005,1)=7005 P(7005,701)=7.6900396347210828241E+2679 P(7005,1401)=1.2659048848290269952E+5322 P(7005,2101)=1.7753130713788487191E+7926 P(7005,2801)=7.2114365718218704695E+10486 P(8005,1)=8005 P(8005,801)=3.9769062582658855959E+3108 P(8005,1601)=4.3272401446508603181E+6174 P(8005,2401)=1.4466844005282015778E+9197 P(8005,3201)=8.3354759867215982278E+12169 P(9005,1)=9005 P(9005,901)=5.6135384805755901099E+3542 P(9005,1801)=1.1194389175552115248E+7038 P(9005,2701)=2.4737530806300682806E+10484 P(9005,3601)=5.6056491332873455398E+13874 P(10005,1)=10005 P(10005,1001)=5.3580936683833889197E+3981 P(10005,2001)=1.3407350644082770778E+7911 P(10005,3001)=1.3407953461588097193E+11786 P(10005,4001)=8.1811569565040010437E+15598 P(11005,1)=11005 P(11005,1101)=1.1340564277915775963E+4425 P(11005,2201)=7.9753039151558717610E+8792 P(11005,3301)=8.0842724022079710248E+13100 P(11005,4401)=2.9749926937463675736E+17340 P(12005,1)=12005 P(12005,1201)=2.1391703159775094656E+4872 P(12005,2401)=3.7994859265471124812E+9682 P(12005,3601)=3.5081603331307865953E+14427 P(12005,4801)=6.9968644993020337359E+19097 P(13005,1)=13005 P(13005,1301)=1.6832659142209713962E+5323 P(13005,2601)=3.1719005022749408296E+10579 P(13005,3901)=1.1206120643200361213E+15765 P(13005,5201)=5.0883658993886790178E+20869 P(14005,1)=14005 P(14005,1401)=2.9074578200382556975E+5777 P(14005,2801)=1.2835011192416517281E+11483 P(14005,4201)=3.8312600202917546343E+17112 P(14005,5601)=8.7456467698123057261E+22654 C(100,1)=100 C(100,21)=2.0418414110621321255E+21 C(100,41)=2.0116440213369968048E+28 C(100,61)=9.0139240300346304925E+27 C(100,81)=1.3234157293921226741E+20 C(200,1)=200 C(200,41)=8.0006165666286406037E+42 C(200,81)=2.4404128184470558197E+57 C(200,121)=1.0891098528606695394E+57 C(200,161)=5.0935602365182339252E+41 C(300,1)=300 C(300,61)=3.5574671252567510894E+64 C(300,121)=3.3878557197772169409E+86 C(300,181)=1.5098730832156289128E+86 C(300,241)=2.2510943427454545270E+63 C(400,1)=400 C(400,81)=1.6703771503944415835E+86 C(400,161)=4.9770797199515347150E+115 C(400,241)=2.2166247162163128334E+115 C(400,321)=1.0537425948749981954E+85 C(500,1)=500 C(500,101)=8.0859177660929770887E+107 C(500,201)=7.5447012685604958486E+144 C(500,301)=3.3587706644089915087E+144 C(500,401)=5.0915068227892187414E+106 C(600,1)=600 C(600,121)=3.9913554739811382543E+129 C(600,241)=1.1667430218545073615E+174 C(600,361)=5.1927067085306791157E+173 C(600,481)=2.5101559893540422495E+128 C(700,1)=700 C(700,141)=1.9971304197729039382E+151 C(700,281)=1.8294137562513979560E+203 C(700,421)=8.1403842518866639077E+202 C(700,561)=1.2548814134936695871E+150 C(800,1)=800 C(800,161)=1.0093242166331589874E+173 C(800,321)=2.8977104736455704539E+232 C(800,481)=1.2892100651978213666E+232 C(800,641)=6.3378002682503352943E+171 C(900,1)=900 C(900,181)=5.1402207737939392620E+194 C(900,361)=4.6256239559539226532E+261 C(900,541)=2.0577328996911473555E+261 C(900,721)=3.2260054093505652100E+193 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
Ruby
Float calculation as Tcl. <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 </lang> 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) <lang ruby>(1..60).to_a.combination(53).size #=> 386206920</lang>
Scheme
<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) </lang>
- Output:
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
Sidef
<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 - e*10.log), e) }
}
func lstirling(n) {
n < 10 ? (lstirling(n+1) - log(n+1)) : (0.5*log(2*Num.pi*n) + n*log(n/Num.e + 1/(12*Num.e*n)))
}
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)}"
}</lang>
- Output:
=> 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
Stata
The comb function is builtin. Here is an implementation, together with perm:
<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))) }</lang>
Swift
Using AttaSwift's BigInt
<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)")
}</lang>
- Output:
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)
Tcl
Tcl doesn't allow the definition of new infix operators, so we define and as ordinary functions. There are no problems with loss of significance though: Tcl has supported arbitrary precision integer arithmetic since 8.5.
<lang tcl># Exact integer versions proc tcl::mathfunc::P {n k} {
set t 1 for {set i $n} {$i > $n-$k} {incr i -1} {
set t [expr {$t * $i}]
} return $t
} proc tcl::mathfunc::C {n k} {
set t [P $n $k] for {set i $k} {$i > 1} {incr i -1} {
set t [expr {$t / $i}]
} return $t
}
- Floating point versions using the Gamma function
package require math proc tcl::mathfunc::lnGamma n {math::ln_Gamma $n} proc tcl::mathfunc::fP {n k} {
expr {exp(lnGamma($n+1) - lnGamma($n-$k+1))}
} proc tcl::mathfunc::fC {n k} {
expr {exp(lnGamma($n+1) - lnGamma($n-$k+1) - lnGamma($k+1))}
}</lang> Demonstrating: <lang tcl># Using the exact integer versions puts "A sample of Permutations from 1 to 12:" for {set i 4} {$i <= 12} {incr i} {
set ii [expr {$i - 2}] set iii [expr {$i - int(sqrt($i))}] puts "$i P $ii = [expr {P($i,$ii)}], $i P $iii = [expr {P($i,$iii)}]"
} puts "A sample of Combinations from 10 to 60:" for {set i 10} {$i <= 60} {incr i 10} {
set ii [expr {$i - 2}] set iii [expr {$i - int(sqrt($i))}] puts "$i C $ii = [expr {C($i,$ii)}], $i C $iii = [expr {C($i,$iii)}]"
}
- Using the approximate floating point versions
puts "A sample of Permutations from 5 to 15000:" for {set i 5} {$i <= 150} {incr i 10} {
set ii [expr {$i - 2}] set iii [expr {$i - int(sqrt($i))}] puts "$i P $ii = [expr {fP($i,$ii)}], $i P $iii = [expr {fP($i,$iii)}]"
} puts "A sample of Combinations from 100 to 1000:" for {set i 100} {$i <= 1000} {incr i 100} {
set ii [expr {$i - 2}] set iii [expr {$i - int(sqrt($i))}] puts "$i C $ii = [expr {fC($i,$ii)}], $i C $iii = [expr {fC($i,$iii)}]"
}</lang>
- Output:
A sample of Permutations from 1 to 12: 4 P 2 = 12, 4 P 2 = 12 5 P 3 = 60, 5 P 3 = 60 6 P 4 = 360, 6 P 4 = 360 7 P 5 = 2520, 7 P 5 = 2520 8 P 6 = 20160, 8 P 6 = 20160 9 P 7 = 181440, 9 P 6 = 60480 10 P 8 = 1814400, 10 P 7 = 604800 11 P 9 = 19958400, 11 P 8 = 6652800 12 P 10 = 239500800, 12 P 9 = 79833600 A sample of Combinations from 10 to 60: 10 C 8 = 45, 10 C 7 = 120 20 C 18 = 190, 20 C 16 = 4845 30 C 28 = 435, 30 C 25 = 142506 40 C 38 = 780, 40 C 34 = 3838380 50 C 48 = 1225, 50 C 43 = 99884400 60 C 58 = 1770, 60 C 53 = 386206920 A sample of Permutations from 5 to 15000: 5 P 3 = 59.9999999964319, 5 P 3 = 59.9999999964319 15 P 13 = 653837183936.7548, 15 P 12 = 217945727984.54794 25 P 23 = 7.755605021026223e+24, 25 P 20 = 1.2926008369145724e+23 35 P 33 = 5.166573982873315e+39, 35 P 30 = 8.610956638634269e+37 45 P 43 = 5.981111043018166e+55, 45 P 39 = 1.6614197342883882e+53 55 P 53 = 6.348201676661335e+72, 55 P 48 = 2.5191276496660396e+69 65 P 63 = 4.123825295988996e+90, 65 P 57 = 2.0455482620718488e+86 75 P 73 = 1.2404570405684596e+109, 75 P 67 = 6.153060717624475e+104 85 P 83 = 1.4085520572027225e+128, 85 P 76 = 7.763183737477006e+122 95 P 93 = 5.164989244208789e+147, 95 P 86 = 2.846665148075141e+142 105 P 103 = 5.406983791334563e+167, 105 P 95 = 2.980039567808848e+161 115 P 113 = 1.462546846791721e+188, 115 P 105 = 8.060774068156828e+181 125 P 123 = 9.413385884788385e+208, 125 P 114 = 4.716503269639238e+201 135 P 133 = 1.345236353714729e+230, 135 P 124 = 6.74020138809567e+222 145 P 143 = 4.0239630289197437e+251, 145 P 133 = 1.6801459658196038e+243 A sample of Combinations from 100 to 1000: 100 C 98 = 4950.000000564707, 100 C 90 = 17310309460118.861 200 C 198 = 19900.000002250566, 200 C 186 = 1.1797916416885855e+21 300 C 298 = 44850.00000506082, 300 C 283 = 2.287708142503998e+27 400 C 398 = 79800.00000901309, 400 C 380 = 2.788360984244711e+33 500 C 498 = 124750.00001405331, 500 C 478 = 1.327364247175741e+38 600 C 598 = 179700.00002031153, 600 C 576 = 4.7916866834178515e+42 700 C 698 = 244650.00002750417, 700 C 674 = 1.454786513417567e+47 800 C 798 = 319600.0000360682, 800 C 772 = 3.933526871778561e+51 900 C 898 = 404550.0000452471, 900 C 870 = 9.803348169192494e+55 1000 C 998 = 499500.0000564987, 1000 C 969 = 7.602322409167201e+58
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 expr C(1000,500)
takes approximately 1000 times longer to compute than expr fC(1000,500)
VBScript
<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=z*i next 'i perm=z end function 'perm
function fact(x) dim i,z z=1 for i=2 to x z=z*i 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</lang>
- Output:
-- 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
Visual Basic .NET
<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</lang>
- Output:
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
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