Date: Feb 5, 2013 3:46 AM
Author: quasi
Subject: Re: how many ways to pick two teams of 10 out of a group of 20 people

G Patel wrote:

>Is it 20 choose 10 or do I have to divide by 2?

You have to divide by 2.

Let S be a set of 20 people. There are (20-choose-10) 10-person
subsets of S, and for each such choice, we can create the teams
A,B where A is the chosen subset and B is the complement of A.
But B is also one of the possible choices for a 10-person subset
of S, and had B been the chosen subset, the teams would also be
A,B. Thus, the correspondence between the set of 10-person
subsets of S and unordered pairs A,B of 10-person teams is a
2-to-1 correspondence. Therefore the number of ways to pick two
10-person teams from 20 people is (20-choose-10)/2.