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# combinations/permutations

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b) How many five-card poker hands containing three of a kind are possible? (Consider a full house to be a three of a kind.)

out of a standard deck of cards.

Guest Sep 16, 2017
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Number of full house hands

We want to choose any 1 of 13 ranks  and and any 3 of 4 cards witihin the chosen rank

Then...we want to choose  two cards from any one of the 12 remaining ranks

So.....full house hands =

C (13 , 1) * C(4,3) * C(12, 1) * C(4,2)  = 3744

Three of a kind  [not counting full house hands ]

We want to choose any 1 of 13 ranks  and and any 3 of 4 cards witihin the chosen rank

Then...we want to choose any two of the 12 remaining ranks with one card from each rank

So...we have

C (13, 1)  * C (4,3)  * C (12,2)  * [C(4,1)]^2  = 54,912

So....the total of three-of-a-kind hands  [ including full house hands ]  =  3744 + 54,912   = 58,656

CPhill  Sep 16, 2017
edited by CPhill  Sep 16, 2017

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