A variant of this problem came up in discussion recently, that I think lends itself to attack via either mathematical “pencil and paper” or “write a program” analysis:

What is the median poker hand? More precisely, among all possible five-card poker hands, what hand ranks higher than exactly half of them?

Of course, the solution is possibly not quite unique, since the ranking of hands is a weak order: suits never decide the relative ranking of hands. For example, there are four possible royal flushes, and no one royal flush beats another. So, as a follow-up combinatorics problem: how many “distinct” hand ranks are there? That is, how many equivalence classes are there in the incomparability relation “hand ties with hand ” on the set of possible hands?