[Math] Prove $\binom{n}{2k+1}=\sum_{i=1}^n{\binom{i-1}{k}\binom{n-i}{k}}$

Question

I have to prove the equality
$$\binom{n}{2k+1}=\sum_{i=1}^n{\binom{i-1}{k}\binom{n-i}{k}}$$

What I can see is that the left hand side is the number of ways to choose $2k+1$ elements from $n$ elements, while the right hand side is the sum of the ways you can choose k elements from one set, and k from another, effectively choosing $2k$ elements from a set of $n-1$ elements.

This doesn't make sense to me. I would expect that the left hand side would be the sum of the ways to choose $2k+1$ elements from $n$ elements, choosing from two different sets.

Can anyone help me with this?

Answer 1

Hint: $$\sum_{i=1}^n{\binom{i-1}{k}\binom{n-i}{k}}=\sum_{i=1}^n{\binom{i-1}{k}\binom{n-i}{k}}\binom{1}{1}$$