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.
+130116
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
