Combinatorial argument to prove counting.

Question

Find a combinatorial argument to prove that for all positive integers $k \leq n$,$${k\choose 2} + {n - k \choose 2} + k(n - k) = {n \choose 2}$$

I'm recently new learning combinatorics and having trouble understand the intuition behind this. I don't either know what's combinatorial argument, but it looks like I have to show the LHS counting is equal to RHS by some tricks. Any suggestion ?

Answer 1

Split a set with $n$ elements into a set with $k$ elements and a set with $n-k$ elements. To choose two elements from the larger set, you can either choose two from the set with $k$ elements, choose two from the set with $n-k$ elements, or choose one from each.