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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.)

 Jan 5, 2020
 #1
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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.

 Jan 5, 2020
 #2
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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:  13C2  =  78

Choose two of the 4 different suits for the first pair:  4C2  =  6

Choose two of the 4 different suits for the second pair:  4C2  =  6

Choose one of the 44 cards that remain:  44

 

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

 Jan 5, 2020

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