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?
 A: The question is not 100% clearly phrased. You could also consider your answer correct.
Say there is a red and a blue team. Then you correctly figured that there are $\binom{30}{10}\binom{20}{10}$ ways to pick 10 players for 10 red and 10 players for the blue team.
But now you can argue that swapping the team colors does not change anything. So if you do not care which team is red and which is blue but only which players play together the number of possibilities is reduced in half.
A: 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.
