Show that $\binom{2n}{2} = 2 \binom{n}{2}+n^2$.
LHS: This is the number of pairs of $2n$ distinct elements.
RHS: We can rewrite this as $2 \binom{n}{2}+ \binom{n}{1} \binom{n}{1}$. So you can divide up the $2n$ distinct elements into two subsets of $n$ elements. Then pick an element from each of these subsets to form a pair. Or you can just choose $2$ elements from the $n$ element subset (and multiply by $2$ since order matters).
Is this the correct idea?