I'm trying to prove that $\sum\limits\_{k=2}^n {k\choose2} = {{n+1}\choose 3}$ for integers $n\geq 2$.

I figured induction was the way to go, so I tried. This is what I've accomplished so far:

Proved it holds for $n=2$.

Assumed it holds for some $n = q$

Used the assumption to get the following:

$$\sum\limits\_{k=2}^{q+1}{k\choose2} = {{q+1}\choose 3}+{{q+1}\choose 2}$$

My goal is to prove that this equals $\displaystyle{{q+2}\choose 3}$ but I can't quite get there. I've expanded the binomial coefficients, but it's a mess.

Looking for help on completing the proof. Of course, in the name of curiosity, other neat methods are also welcome!