combinatorics - Proof of a Binomial Identity using a combinatorial argument

Question

Prove that if $k$ and $l$ are two positive integer with $k ≥ l$, then $\binom{2k}{2} =\binom{k−l}{2}+ \binom{k+l}{2}+ k^2 − l^2$
using a combinatorial argument.

I tried using Vandermonde's Identity and Pascal's Identity but that's leading me nowhere.
Any lead is very welcome.

Thanks

Answer

You want a combinatorial argument...

Say you have $2k$ items and you want to select $2$ of them. Divide the $2k$ items into two groups, one with $k+\ell$ items and one with $k-\ell$ items.

To select two items out of the overall $2k$, you could select two from the group of $k+\ell$ items; or two from the group of $k-\ell$ items; or you could select one from the group of $k+\ell$ and one from the group of $k-\ell$ items. If you add all three possibilities, you should get $\binom{2k}{2}$.

How many ways to pick two from $k+\ell$? How many ways to pick two from $k-\ell$? How many ways to pick one from $k+\ell$ and one from $k-\ell$?