Combinations and permutations: Difference between revisions
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Printing of test cases is performed incrementally, which is associated with the characteristics of the device output. |
Printing of test cases is performed incrementally, which is associated with the characteristics of the device output. |
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=={{header|Perl}}== |
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Although perl can handle arbitrarily large numbers using Math::BigInt and Math::BigFloat, it's |
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native integers and floats are limited to what the computer's native types can handle. |
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As with the perl6 code, some special handling was done for those values which would have overflowed the native floating point type. |
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<lang perl>use strict; |
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use warnings; |
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showoff( "Permutations", \&P, "P", 1 .. 12 ); |
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showoff( "Combinations", \&C, "C", map $_*10, 1..6 ); |
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showoff( "Permutations", \&P_big, "P", 5, 50, 500, 1000, 5000, 15000 ); |
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showoff( "Combinations", \&C_big, "C", map $_*100, 1..10 ); |
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sub showoff { |
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my ($text, $code, $fname, @n) = @_; |
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print "\nA sample of $text from $n[0] to $n[-1]\n"; |
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for my $n ( @n ) { |
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my $k = int( $n / 3 ); |
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print $n, " $fname $k = ", $code->($n, $k), "\n"; |
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} |
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} |
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sub P { |
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my ($n, $k) = @_; |
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my $x = 1; |
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$x *= $_ for $n - $k + 1 .. $n ; |
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$x; |
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} |
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sub P_big { |
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my ($n, $k) = @_; |
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my $x = 0; |
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$x += log($_) for $n - $k + 1 .. $n ; |
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eshow($x); |
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} |
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sub C { |
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my ($n, $k) = @_; |
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my $x = 1; |
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$x *= ($n - $_ + 1) / $_ for 1 .. $k; |
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$x; |
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} |
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sub C_big { |
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my ($n, $k) = @_; |
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my $x = 0; |
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$x += log($n - $_ + 1) - log($_) for 1 .. $k; |
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exp($x); |
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} |
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sub eshow { |
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my ($x) = @_; |
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my $e = int( $x / log(10) ); |
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sprintf "%.8Fe%+d", exp($x - $e * log(10)), $e; |
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} |
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</lang> |
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Since the output is almost the same as perl6's, and this is only a Draft RosettaCode task, I'm not going to bother including the output of the program. |
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=={{header|Perl 6}}== |
=={{header|Perl 6}}== |
Revision as of 02:14, 21 January 2014
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) |
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:
- 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.
This 'floating point' code could be implemented using an approximation, e.g., by calling the Gamma function.
See Also:
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
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>
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.
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 perl6 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)), $e; } </lang>
Since the output is almost the same as perl6's, and this is only a Draft RosettaCode task, I'm not going to bother including the output of the program.
Perl 6
Perl 6 is fairly limited with very large floating values. It tends to consider them as infinite when they reach or so. This forces us to use only logarithms and to tweak them in order to show the exponential.
Notice that Perl6 can process arbitrary long integers, though. So it's not clear whether using floating points is useful in this case.
<lang perl6>multi P($n, $k) { [*] $n - $k + 1 .. $n } multi C($n, $k) { P($n, $k) / [*] 1 .. $k }
sub lgamma(\z) {
z < 10 ?? lgamma(z+1) - log(z) !! .5*log(2*pi*z)+ z*log(z/e+1/(12*e*z))- log(z)
}
role Logarithm {
method gist {
my $e = (self/10.log).Int; sprintf "%.8fE%+d", exp(self - $e*10.log), $e;
}
} multi P($n, $k, :$float!) {
(lgamma($n+1) - lgamma($n -$k +1)) but Logarithm
} multi C($n, $k, :$float!) {
(lgamma($n+1) - lgamma($n -$k +1) - lgamma($k+1)) 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
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...)
REXX
The hard part of this REXX program was coding the DO loops for the various ranges. <lang rexx>/*REXX program to compute a sampling of combinations and permutations. */ numeric digits 100 /*use hundred digits of precision*/
do j=1 to 12 /*show permutations from 1──► 12 */ _=; do k=1 to j /*step through all J permutations*/ _=_ 'P('j","k')='perm(j,k)" " /*add an extra blank between #s. */ end /*k*/ say strip(_) /*show a horizontal line of PERMs*/ end /*j*/
say
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 #s. */ end /*k*/ say strip(_) /*show a horizontal line of COMBs*/ end /*j*/
say numeric digits 20 /*force floating point for big #s*/
do j=5 to 15000 by 1000 /*show a few permutations, big #s*/ _=; do k=1 to j by j%10 for 5 /*go through some J permutations.*/ _=_ 'P('j","k')='perm(j,k)" " /*add an extra blank between #s. */ end /*k*/ say strip(_) /*show a horizontal line of PERMs*/ end /*j*/
say
do j=100 to 1000 by 100 /*show a few combinations, big #s*/ _=; do k=1 to j by j%5 /*step through some combinations.*/ _=_ 'C('j","k')='comb(j,k)" " /*add an extra blank between #s. */ end /*k*/ say strip(_) /*show a horizontal line of COMBs*/ end /*j*/
exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────COMB subroutine─────────────────────*/ comb: procedure; parse arg x,y /*args: X things, Y at-a-time.*/ if y>x then return 0 /*oops-say, to big a chunk. */ if x=y then return 1 /* X things same as chunk size. */ if x-y<y then y=x-y /*switch things around for speed.*/ call .cmbPrm /*call sub to do heavy lifting. */ return _/!(y) /*perform one last division. */ /*──────────────────────────────────PERM subroutine─────────────────────*/ perm: procedure; parse arg x,y; call .cmbPrm; return _ /*──────────────────────────────────.CMBPRM sugroutine──────────────────*/ .cmbPrm: _=1; do j=x-y+1 to x; _=_*j; end; return _ /*──────────────────────────────────! subroutine────────────────────────*/ !: procedure; parse arg x; !=1; do j=2 to x; !=!*j; end; return !</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(1, :*) end
def combination(k) self.permutation(k) / (2 .. k).inject(1, :*) 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
- Integer has no maximum:
p 15000.permutation(74) #=> 896237613852967826239917238565433149353074416025197784301593335243699358040738127950872384197159884905490054194835376498534786047382445592358843238688903318467070575184552953997615178973027752714539513893159815472948987921587671399790410958903188816684444202526779550201576117111844818124800000000000000000000 </lang>
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)