combinatorially prove that for all positive integers $k \le n$

Question

$\binom{k}{2} + \binom{n-k}{2} + k(n-k) = \binom{n}{2}$

I can prove it algebraically.

Answer 1

Hint: Consider picking 2 people from a group of $k$ men and $n-k$ women. You can pick 2 men, 2 women, or one of each...

Answer 2

Hint:

We can rewrite it as $$\binom{k}{2} + \binom{n-k}{2} + \binom{k}{1}\binom{n-k}{1} = \binom{n}{2}$$