Giving an proof on a combinatorial statement

Question

Prove with a combinatorial argument that $\displaystyle\binom{a+b}{2}-\binom{a}{2}-\binom{b}{2}=ab.$

I'm assuming we can give a committee forming argument, but I'm not sure how to start.

Answer 1

Rewrite as $$\binom{a+b}{2}=\binom{a}{2}\binom{b}{0}+\binom{a}{1}\binom{b}{1}+\binom{a}{0}\binom{b}{2}$$ and note that both sides count the number of ways to choose a pair of people from $a$ men and $b$ women. The left hand side is clear. The right hand side performs the count according to three cases:

• $2$ men and $0$ women
• $1$ man and $1$ woman
• $0$ men and $2$ women

Answer 2

HINT: It’s easier to prove that $\binom{a+b}2=ab+\binom{a}2+\binom{b}2$. You have $a$ men and $b$ women, and you want to choose $2$ of them to represent the whole group. Find two different ways to compute the number of ways to choose the two representatives if their sexes don’t matter.