This problem is an example in a textbook. I don't fully understand the result.
In order to play a game of basketball, 10 children at a playground divide themselves into two teams of 5 each. How many different divisions are possible?
I answer it as follows: $$ {10 \choose 5}{5\choose5} = {10\choose5} $$
My logic for this solution is: you have ten players, choose 5 to form a team. (Note you have de facto created the second team, as the resulting 5 that were not chosen in any given combination comprise the competing team). For consistency, 5 players remain choose 5.
The answer the book gives is 126. I'm not sure what is wrong with my logic.
You're treating the teams as labelled; the book is treating the teams as unlabelled, i.e. it doesn't matter whether you select a team or its complement, you still get the same two teams, so you need to divide by $2$.