In a play, 10 actors (5 males and 5 females) are to be selected for 3 male roles and 3 female roles. In how many ways can this distribution happen? To solve that, I'm thinking of the rule of product. For each role, 5 different actors could be selected, so I should consider (5 * 5 * 5) + (5 * 5 * 5) = 250 as the answer?

You should consider the number of ways to select the male roles, given by the coefficient $\binom{5}{3}$, which is the same for the female. Now, by the rule of product, the number of ways to complete the task is $\binom{5}{3}\binom{5}{3}=100$.
