[Math] How many ways are there of splitting twelve people into two groups of the same size

Question

Twelve people need to be split up into teams for a quiz. How many ways are there of splitting them into two groups of the same size?

I did $12 C 6$, which gives $924$, however the answer is $\frac{1}{2}(12 C 6)$, which is $462$. Why is the $12 C 6$ halved?

Any help will be much appreciated, thanks in advance.

Answer 1

$12\choose 6$ double counts. For instance you could choose A, B, C, D, E, F for one group, leaving G, H, I, J, K, L for the other; or you could choose G, H, I, J, K, L for one group, leaving A, B, C, D, E. F. But this actually gives you the same two groupings.