I think its something like that: 
$cards -= $holes; # one card per hole is required,
                  # so let's disregard those right away
my $r = $cards;
my $n = $cards + $holes - 1;

my $result =
  factorial($n) / ( factorial($r) * factorial($n - $r) );

sub factorial {
    use List::Util 'reduce';

    return 1 if $_[0] <= 0;
    return reduce { $a * $b } 1 .. $_[0];
}

[download]
If that's correct, i'll have more time for explanations later :)

In reply to Re: Combinatorics problem. by Anonymous Monk
in thread Combinatorics problem. (Updated with more info.) by BrowserUk

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.