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.