In poker, a hand with 5 cards is said to have two pairs if it contains two cards of one rank, two cards of another rank, and a fifth card of a third rank. (For example, the hand JJ227 has two pairs.) How many different hands with 5 cards have two pairs? (Assume that the cards are taken from a standard deck of 52 cards, and that the order of the cards does not matter.)

Guest Jan 5, 2020

#1**+1 **

There are 13*3 possibilities for the first pair, leaving 12*3 possibilities for the 2nd pair, leaving 11*4 possibilities for the singleton. So: 13*3*12*3*11*4 = 61776 in total.

Guest Jan 5, 2020

#2**+1 **

I have a different way of looking at this problem (that results in a different answer):

Choose two of the 13 different ranks to get the two pairs: _{13}C_{2} = 78

Choose two of the 4 different suits for the first pair: _{4}C_{2} = 6

Choose two of the 4 different suits for the second pair: _{4}C_{2} = 6

Choose one of the 44 cards that remain: 44

Multiply these answers together: 78 x 6 x 6 x 44 = 123 552

geno3141 Jan 5, 2020