I've looked at permutations and combinations, and have decided they're not what I'm looking for. I have a game at home called Six Cubes. It's a dice game -- you roll six dice, and if specific combinations come up, you can store them away for points, and decide to roll the remaining dice or stop. The dice are your standard d6 (that's a gaming term).

I'm trying to calculate the number of rolls possible. Everything from "1 1 1 1 1 1" to "6 6 6 6 6 6". However, the roll "1 1 1 1 1 2" is no different from "1 1 1 2 1 1". I'm trying to determine the mathematical approach to calculating the number of rolls possible. I can produce them (there are 462), but I can't figure out a generic mathematical formula to get that number.

From a regex standpoint, it's /^(?=.{6}$)1*2*3*4*5*6*/. (This is extraneous, but it defines the strings that match. "111112" is valid, "121111" is not,, because it's "identical".)

Here's my Perl code. It looks eerily familiar, like I've seen this somewhere before, but I don't know where offhand: 

sub unordered_sets {
  my ($in, $out, $left, $idx, $curr) = @_;
  push @$out, [@$curr] and return if $left == 0;
  $idx ||= 0;
  for ($idx .. $#$in) {
    push @$curr, $in->[$_];
    unordered_sets($in, $out, $left-1, $_, $curr);
    pop @$curr;
  }
}

my @sets;
unordered_sets [1..6], \@sets, 6;

[download]
_____________________________________________________
Jeff japhy Pinyan, P.L., P.M., P.O.D, X.S.: Perl, regex, and perl hacker
How can we ever be the sold short or the cheated, we who for every service have long ago been overpaid? ~~ Meister Eckhart

In reply to unordered sets of N elements by japhy

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.