I've got a good handle on how to prove binomial identities using a committee argument, but how would I go about trying to use the Binomial Theorem to prove them?
For example, $\dbinom{2n}2=2\dbinom n2+n^2$
I've got a pretty good handle on proving this identity using a committee argument where I form a committee of size 2 from an initial group size of 2n, and then I break up this committee into two separate groups.
My question is, how can I use the Binomial Theorem to prove this identity?