I place $n$ pairs of socks (thus $2 n$ socks) in a line in such a way that the left sock is to the right of the right sock for each pair. How many different ways can I place my socks like this?
If we place the $2n$ socks in a line without worrying about the condition (the left sock of a pair is always to the right of the other sock of the pair), we obtain a permutation of the $2n$ socks, so we have $(2n)!$ possibilities. Once we have done this, we can "correct" the permutation: it is only necessary to possibly swap the positions of the left and right socks of the same pair to obtain a configuration satisfying the condition.

However, reasoning this way, we have counted the same configuration multiple times: indeed, $2^n$ permutations give the same configuration after correction, because we have $n$ pairs that may or may not have been swapped. Thus, the number of such configurations is $\frac{(2n)!}{2^n}$.