Question 1022434: Find the number of full houses in a poker hand, that is, the number of poker hands with three of a kind and two of a kind.
Answer by mathmate(429)   (Show Source):
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Question:
Find the number of full houses in a poker hand, that is, the number of poker hands with three of a kind and two of a kind.
Solution:
We assume the question applies to a single full deck of 52 cards.
From the four cards in a rank (Ace, King, ...threes, twos), there are C(4,3)=4 ways to form 3 of a kind, and C(4,2)=6 ways to form two of a kind, where
C(n,r)=n!/(r!(n-r)!).
Furthermore, there are 13 ways to choose three of a kind, and 12 ways to choose two of a kind (or vice versa, the result is the same).
So the total number of full houses is 4*6*13*12=3744.
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