# How to prove that $\binom{n}{2k+1}=\sum_{i=k+1}^{n-k}\binom{i-1}{k}\binom{n-i}{k}$?

I've tried to prove that $\binom{n}{2k+1}=\sum_{i=k+1}^{n-k}\binom{i-1}{k}\binom{n-i}{k}$ by using combinatorial proof.
The LHS is the number of binary vectors of size $n$ with $2k+1$ zeros. I can't find an explanation regarding the RHS.

Consider a binary vector of size $n$ with $2k+1$ zeroes. The $(k+1)$'th zero can happen anywhere between the $k+1$ and $n-k$ positions, hence the summation.
Let the $k$'th zero be on the $i$'th position. The two factors $\binom{i-1}k$ and $\binom{n-i}k$ respectively denotes how many ways to assign $k$ zeroes before and after the $k$'th zero. The $k$ zeroes before and after the centre zero can be arranged independently, hence the product.