Ok, this is what I need to do:

"Prove that for any n and r where 0 <= r <= n,

C(n,2) = C(r,2) + C(n-r,2) + r(n-r)"

I can plug in numbers and see that this seems to hold true. But I am not sure how to prove this for any n and r that meets the condition.

As far as an actual proof goes, I don't even know how to approach this. Can anyone give me a suggestion or poke me in the right direction?