Show that $\binom{2n}{n}$ is an even number, for positive integers $n$. I would appreciate if somebody could help me with the following problem

Show by a combinatorial proof that
$$\dbinom{2n}{n}$$
is an even number, where $n$ is a positive integer.

I tried to solve this problem but I can't.

Let $S$ be all subsets of $T=\{1,2,3,\dots,2n\}$ of size $n$. There is an equivalence relation on $S$ where every equivalence class has two elements, $\{A,T\setminus A\}$.
