discrete and cominatorial math chapter 1_2

#/usr/bin/env perl6
use v6;


sub postfix:<!>(Int $N where $N > 0) is tighter(&infix:</>) {
	factorial( $N )
}


"counting arrangement by factorial".say;

say q:to/EOF/;
EX:
	10!/5! = ?
EOF

say 10! / 5!;

"counting permutation".say;

say q:to/EOF/;
EX:
	P( 7 , 2 )
EOF
say "P( 7 , 2 ) = ",count( ^7 ):p(2) ; 


say q:to/EOF/;
EX:
	counting the arrangement of MASSASAUGA
EOF
say "10! / 4! * 3! = ",  count('MASSASAUGA'.split('',:skip-empty), :a) ; 

say q:to/EOF/;
EX:
	There are 6 people at a party sitting at a round table with 6 seats: 
	How many ways can the 6 people be seated?
EOF
say "we have ", count(^6):cp , " way to sit";


say q:to/EOF/;
EX:
	There are 6 people at a party sitting at a round table
	with 6 seats: A, B, C, D, E and F. A CANNOT sit next to either D or F. 
	How many ways can the 6 people be seated?
	need use Inclusion-exclusion principle
EOF

say "we have ",
	count(<A B C D E F>,:cp) 
	- ( count(<AD B C E F>,:cp) * 2! + count(<AF B C D E>, :cp) * 2! ) 
	+ count(<DAF B C E>, :cp) * 2!
	, " way to sit";

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

# p flag = permutation 
multi sub count(@N where @N.elems > 0, Int :$p! where $p < @N.elems --> Int:D) {
	Int( @N.elems! / (@N.elems - $p )! ) 
}

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

# cp flag  = circular permutation
multi sub count(@N where @N.elems > 0, Bool :$cp! --> Int:D) {
	Int( (@N.elems - 1)! )
}