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?

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.
