discrete and cominatorial math chapter 1_3

#/usr/bin/env perl6
use v6.c;


sub postfix:<!>(Int $n) is tighter(&infix:</>) {
	factorial( $n )
}

multi sub factorial( Int $n where $n > 0 ) {
    [*] 1..$n
}

multi sub factorial( Int $n where $n == 0 ) {
	1
}

multi sub combin(@n where @n.elems > 0, Int $r where $r < @n.elems) {
	@n.elems! /( $r! * ( @n.elems - $r )! ) 
}

multi sub combin(@n where @n.elems > 0, Int $r where $r == @n.elems) {
	return 1
}

multi sub permu(@n where @n.elems > 0, Int $r where $r < @n.elems --> Int:D) {
	Int( @n.elems! / (@n.elems - $r )! ) 
}

multi sub permu(@n where @n.elems > 0, Bool :$a! --> Int:D) {
	Int( @n.elems! / [*] @n.Bag.grep({ $_.value !== 1 }).map({ $_.value! }) ) ;
}


multi sub binomial_coefficient(@n, @t, Int :$exponet! where $exponet == 0  --> Real:D) {
	1
}

multi sub binomial_coefficient(@n, @t, Int :$exponet! 
		where { [&&] (
				@n.elems == @t.elems == 2,
				@t »<=» $exponet,
				$exponet > 0, 
				|@n »~~» Real, 
				|@t »~~» Int) }
		--> Real:D) {
	my $k = $exponet - @t[0];
	combin(^$exponet,	$k) * ( @n[0] ** @t[0] ) * ( @n[1] ** @t[1] ) 
}

multi sub multinomial_coefficient(@n, @t, Int :$exponet! where $exponet == 0  --> Real:D) {
	1
}

multi sub multinomial_coefficient(@n, @t, Int :$exponet! 
		where { [&&] (
				@n.elems == @t.elems > 2,
				@t »<=» $exponet,
				$exponet > 0, 
				|@n »~~» Real, 
				|@t »~~» Int)  }
		--> Real:D) {
	$exponet! / ( [*] @t»! ) 
		* ( [*] ( @n Z** @t ) )
}

my $ans1 = combin(^10, 6);
my $ans2 = 	 permu('TALLAHASSEE'.comb.grep({$_ ne 'A'})):a
			 * combin( ^('TALLAHASSEE'.comb.grep({$_ ne 'A'}).elems + 1),
				 'TALLAHASSEE'.comb.grep('A').elems);
my $ans3 = binomial_coefficient((1,2), (2,5), :exponet(7));
my $ans4 = multinomial_coefficient((2,3,4), (2,5,3), :exponet(6) );

say qq:to/EOF/;
EX:
	c(10, 6) = ?
sol: {$ans1}
EX:
	TALLAHASSEE 這些安排方法中有 A不相鄰多少種	
sol: {$ans2}
EX:
	A:	(x + 2y)⁷  => x²y⁵ 的係數為
	B:	(2x + 3y + 4z)⁶  => x²y⁵z³ 
sol:
	A = {$ans3}
	B = {$ans4}
EOF