A standard deck of 52 playing cards consists of 4 suits () each with 13 ranks (A, 2,3, ..., J,K,Q). A hand consists of 5 cards in the common poker games.
How many hands are possible?
How many 4 of a kind hands are there? (4 cards of the same rank and 1 other card)
How many full house hands are there? (3 cards of the same rank, and 2 cards of the same (but different) rank)
Note to poker players. Since Full House Hands are 6 times more plentiful than 4 of a Kind Hands, we see why a 4 of a Kind beats a Full House.
- Number of Hands:
- Since there are 52 cards and a hand consists of any 5 of them, we must count the number of ways to select 5 from a set of 52 (order doesn't matter), and this is C(52,5) = 52!/(5! 47!) = 52(51)(50)(49)(48)/5(4)(3)(2)(1) = 2,598,960.
- 4 of a Kind Hands:
- We count the number of these hands in a two step process. First we pick the rank of the 4 cards (and this effectively selects the four cards since we must use all of them and there is only one way to do that), and then we select the remaining card. In
the first step, we choose 1 out of 13 ranks, 13 ways to do this. In the second step, we choose 1 out of 48 remaining cards (52 - 4 selected in the first step). By the multiplication principle, there are 13(48) = 624 hands of this type.
- Full House Hands:
- We again use a multi-step process to count these hands. First we pick the rank of the 3 cards with the same rank, then we select the 3 cards of this rank, next we pick the rank of remaining 2 cards and finally we select the 2 cards of that rank. There
are 13 ways to pick the first rank, C(4,3) = 4 ways to select the 3 cards of that rank, 12 ( = 13-1 since we can not pick the same rank a second time) ways to pick the second rank, and C(4,2) = 6 ways to select the 2 cards of that rank. So, the multiplic
ation principle says that there are 13(4)(12)(6) = 3,744 hands of this type.