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My question is with regards to combinations and permutations. How many ways are there to select n unique objects from x number of identical object pairs?

To make this question clearer, here is a practical scenario: find the number of ways of forming a group of 10 people from 15 couples if the group contains no couples.

My initial solution is this. First, select 10 couples out of 15 couples. Then, from the 10 couples, select another 5 twice. From each of this five couples select one person.

The problem is (or at least my problem is) that I don't know how to put this process into permutations and combinations language. Any advice/help is much appreciated! Thanks!

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Each person selected represents one of the couples, so that there are 10 couples represented. $^{15}C_{10}$ ways to chose the couples.

Each couple can be represented by one of two people. $2^{10}$ ways to do this.

In total, $2^{10}.^{15}C_{10}$

= 1024 * 15! / (10! * 5!)

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