Simplify $\binom{m+n}{2} - \binom{m}{2} - \binom{n}{2}$.

Question

Simplify $\binom{m+n}{2} - \binom{m}{2} - \binom{n}{2}$.

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I'm confused on how to approach this problem. I can't think of any counting argument that will help me, and any 1-1 correspondence. All solutions are appreciated.

Answer 1

HINT: You have $m$ men and $n$ women in a room. $\binom{m+n}2$ is the number of ways to pick two people in the room. $\binom{m}2$ is the number of ways to pick two of the men, and $\binom{n}2$ is the number of ways to pick two of the women. If you remove those possibilities, what’s left?

Answer 2

Algebraic approach:
Use ${k \choose 2} = \frac{k(k-1)}{2}$ and simplify.

Combinatorial approach:
Suppose you have $m$ distinct red balls and $n$ distinct green balls. You want to choose $2$ of the $m+n$ total balls but you don't want them to both be red and you don't want them to both be green.