jmc

Question

A play has two different male roles, two different female roles and two different roles that can be either gender. Only a man can be assigned to a male role, and only a woman can be assigned to a female role. If five men and six women audition, in how many ways can the six roles be assigned?

Accepted Answer

We start with the most restrictive conditions: the roles that are not open to both men and women. First, we will fill the two male roles. There are 5 men, so there are 5 4= 20 ways to assign 2 of the 5 men to the distinct male roles. Doing the same for the female roles, there are 6 5 = 30 ways. Finally, of the 5+6 - 2 - 2 = 7 remaining actors, there are 7 6 = 42 ways to assign the leftovers to the either-gender roles. Multiplying, there are 20 30 42 = 25200 ways of assigning the six roles.