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# EXTREMELY CHALLENGING

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In poker, a pair is formed from two cards of the same rank. A hand with  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  cards and get two pairs? (Assume that we are using a standard deck of  cards, and that the order of the cards does not matter.)

Jan 3, 2020

#2
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We have   C(13.2)  ways of choosing the two ranks

And in each of these ranks we have    C(4,2)   ways of choosing the  pair within a  rank

And we  have   C(44,1)  ways of choosing the remaining card

So

C(13,2)  * [ C(4,2) ]^2  * C(44,1)    =

123552 ways   Jan 3, 2020

#1
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there are 52 ways of choosing the first card

there are 3 ways of chooing its pair

there are 48 ways of choosing the next number

and 3 ways of choosing its pair

there are 44 ways of choosing the single card

so that is    52*3*48*3*44 = 988416

Jan 3, 2020
#2
+1

We have   C(13.2)  ways of choosing the two ranks

And in each of these ranks we have    C(4,2)   ways of choosing the  pair within a  rank

And we  have   C(44,1)  ways of choosing the remaining card

So

C(13,2)  * [ C(4,2) ]^2  * C(44,1)    =

123552 ways   CPhill Jan 3, 2020