In a deck of 52 playing cards (jokers not allowed), how many five-card poker hands consisting of three of a kind and a pair (a full house) are possible?

Question

In a deck of 52 playing cards (jokers not allowed), how many five-card poker hands consisting of three of a kind and a pair (a full house) are possible?

Answer 1

There are 13 ranks to choose from, let's say for the 3 of a kind. (Those ranks are A,K,Q,J,10,9,8,7,6,5,4,3,2).

There are 4 cards of each rank.

(₁₃C₁)(₄C₃) gives the number of ways to get 3 cards of the same rank. Now we need to find the number of ways to get the pair (2 cards of a different rank.

There are 12 ranks remaining to choose from:

(₁₂C₁)(₄C₂), but this time we are only choosing two of the 4 from the selected rank.

The total number of full house hands:

(₁₃C₁)(₄C₃)(₁₂C₁)(₄C₂)

(13)(4)(12)(6) = ?