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How many ways can we send 10 distinguishable letters to 5 people such that each person gets exactly 2 letters.

Any help would be appreciated

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Umm, by tearing one of the letters in half? Did you really mean to make the edit that you made earlier? –  user22805 Feb 28 '12 at 6:31
    
@David: I rolled back user324324's inappropriate edit. –  Jonas Meyer Jun 17 '12 at 17:32
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2 Answers

There are $\tbinom{10}2$ ways to choose two of the letters to go to Person 1. Once they’ve been chosen, there are $\tbinom 82$ ways to choose two of the remaining $8$ letters to go to Person 2, so there are $\tbinom{10}2\tbinom 82$ ways to choose the four letters to go to the first two people on the list.

Can you take it from there?

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Put the letters in order, so that the first two letters will go to Alice, the next two will go to Bob, the next two to Charlie and so on. There are 10! ways to order the letters thus.

But this counts each possibility $2^5$ times, since Alice's letters could be placed in either of two orders, so could Bob's, so could Charlie's, and so on.

So the answer is $\frac{10!}{2^5}$ = 113400

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