Combinatorial proof for $\binom{2n}{2} = 2\binom{n}{2} + n^2$

Question

I completely understand the algebra behind this, but I'm having trouble writing a worded combinatorial proof to show $\binom{2n}{2} = 2\binom{n}{2} + n^2$

Can anyone at least hint me in the right direction?

Answer 1

Hint: Divide the set of $2n$ elements into two sets of $n$ each (say $A$ and $B$). Then selecting $2$ elements from the original set is equivalent to one of the following:

1. Select $2$ elements from $A$ or

2. Select $2$ elements from $B$ or

3. Select an element from $A$ and one from $B$.