How many ways to choose teams $30$ people want to play capture the flag. There are two teams. Each team has ten people. How many ways to choose the teams? I figure there are $30\choose 10$$20\choose 10$ ways to choose members for the team. I am told that this is "double counting", and the answer should be half of $30\choose 10$$20\choose 10$Why is this so?

Actually, there are two types of situations


*

*The teams are labelled (distinguishable), e.g. Lions and Tigers
Here you would not divide by $2$,
as being on the Lions team is not the same as being on the Tigers team.

*The teams are unlabelled (indistinguishable)
Here you would divide by $2$, as explained in the various comments.
In the absence of any explicit mention of teams being labelled, we presume that they are unlabelled.
As a safeguard, you could add the above remark to your answer, if remarks are permitted.
