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In poker, a pair is formed from two cards of the same rank. A hand with 5 cards is called two pairs if it has two pairs of cards of different ranks, and a fifth card that does not match either pair. (For example, the hand KK338 has two pairs.) How many ways can you be dealt 5 cards and get two pairs? (Assume that we are using a standard deck of 52 cards, and that the order of the cards does not matter.)

 Dec 31, 2019
 #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.

 Dec 31, 2019
 #2
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We have 2 of 13  ranks from which  to choose the first pair  = C(13,2)

And we  have C (4,2)  ways to choose the cards within each rank  = [ C(4,2) ]^2

 

And we have  44 ways to choose the last card....so

 

C(13,2) * [C(4,2)]^2  * 44  = 

 

123,552  ways

 

 

cool cool cool

 Dec 31, 2019
edited by CPhill  Dec 31, 2019

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