discrete and cominatorial math chapter 1_2
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#/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)! )
}
Raku
INFO