How many different 5card poker hands are fullhouses Recall t

How many different 5-card poker hands are full-houses? Recall that a 52-card deck consists of the cards: A, 2,..., 10, J, Q, K, A,..., K, A,..., K. A full-house is a five-card hand that consists of three cards with the same face-value and an addition pair of cards with the same face-value (e.g. 2, 2, 2, 8, 8). The order of the cards in the hand doesn\'t matter, (e.g. 8, 2, 2, 8, 2 is considered the same hand as the previous example).

Solution

There are 52C5 ways to choose 5 cards from 52. All these ways are equally likely. Now we will count the number of \"full house\" hands.

For a full house, there are (13C1) ways to choose the kind we have three of. For each of these ways, the actual cards can be chosen in (4C3) ways. For each way of getting so far, there are (12C1) ways to choose the kind we have two of, and for each there are (4C2) ways to choose the actual cards. So our probability is

(13C1*4C3*12C1*4C2/52C5)