Mark Gritter (markgritter) wrote,
Mark Gritter

Counting cards

timprov noted that for the purposes of playing Stud, not all 52! deck orderings are distinct. One reason is because the order of your downcards doesn't matter. There are two ways the downcards can be arranged in the deck, per player, so the number of distinct Stud games is at most 52!/16 52!/(2^8).

I can think of two ways to reduce this number further, but they are not as simple. First, suit doesn't matter if the lowest card on 3rd street is unique (no other card of the same rank). In these situations you can pretend all 4! assignments of suits to suits are the same game.

Next, the game is unchanged by rotation (of hands to players) as long as no two players ever have the same board. (Otherwise action depends on the dealer's position.)

A-5 draw lowball (without the peek-and-kill rule) at first looks a bit simpler; there are 5!=120 permutations of each player's hand that are identical, and we can also remap suits at will. But position matters so we can't rotate players. So there should be 52!/(120^6*24) = 1.1*10^54 different games for six players. But, if you are playing triple draw then the discards may get reshuffled!
Tags: lowball, math, poker, stud, theory
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