Just off the top of my head, this makes me think of $c-$p nested loops to pick slots for the surplus cards like: 

my $p = 7;  # Number of pigeon holes.
my $c = 12; # Number of cards
for my $s1 ( 1..$p ) {
    for my $s2 ( $s1..$p ) {
        for my $s3 ( $s2..$p ) {
            ... Need $c-$p loops

[download]

Which translates to simple code if I don't worry about being more efficient, but it also handles your 7,12 "instantly" for me. 

#!/usr/bin/perl -w
use strict;
use Algorithm::Loops 'NestedLoops';

my( $p, $c ) = @ARGV;
my $iter = NestedLoops(
    [ [ 1..$p ], ( sub { [ $_..$p ] } ) x ($c-$p-1) ],
);
my @s;
while( @s = $iter->() ) {
    my @p = (1) x $p;
    $p[$_-1]++ for @s;
    print "@p\n";
}

[download]

- tye        

In reply to Re: Combinatorics problem. (NestedLoops) by tye
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.