Chosing $2$ person from each groups using Product rule

Question

Group A has $10$, Group B has $15$, and Group C has $20$ persons. What if only 2 persons can be chosen and they should be from different groups, what should I do?

So far I can only think of simple rule of product which just multipies things.

Answer 1

You have: $(\binom{10}{2}+\binom{10}{3})(\binom{15}{2}+\binom{15}{3})(\binom{20}{2}+\binom{20}{3})$ ways of choosing $2$ or $3$ persons from each group. If you only want to choose $2$ persons from each group, then you have $\binom{10}{2}\cdot \binom{15}{2}\cdot \binom{20}{2}$ ways.