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
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
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?
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