Prove the identity $\binom{2n}{2}$ = $\binom{n}{2}+\binom{n}{n-2}+n^2$ where $n\geq2$ using a combinatorial proof.

Question

Prove the identity $\binom{2n}{2}$ = $\binom{n}{2}+\binom{n}{n-2}+n^2$, where $n\geq2$, using a combinatorial proof.

I've tried to think of it in terms of a counting problem. I think that for the left hand side that given a group of people of size 2n you'll be choosing 2 people out of that. I think that the right hand side may be breaking down the left hand side into cases, but I'm not sure.

Answer 1

Hint: Split the $2n$ elements into two halves of size $n$. What are the possible ways $2$ chosen elements can be distributed between the halves? How many ways are there to choose them consistently with each distribution?