How many different casts are there?

Question

Assume that there are 6 unfilled roles: 4 male and 2 female. There are 8 men and 4 women, including Jennifer, auditioning for a part in the play. 

(1) How many different casts are there?

(2) How many of these casts include Jennifer?

Answer 1

(1) You can split this into two questions: How many ways are there to choose the female cast? and how many ways to choose the male cast. There are 4 women and 2 female roles, so there are (4 choose 2) = 4!/(2!*(4-2)!) = 4*3/(2*1) = 6 ways of choosing the female cast. There are 8 men and 4 male roles, so there are (8 choose 4) = 8!/(4!*(8-4)!) = 8*7*6*5/(4*3*2*1) = 70 ways of choosing the male cast. For each possible female cast, there are 70 possible male casts, so the total number of casts is 6*70 = 420.

(2) Here, we don't need to recalculate the number of male casts. Since we know Jennifer gets a role, there is only 1 open role left. There are only 3 possible female casts, one for each woman who could take the remaining role. Multiplying the female casts by male casts to get the total number of casts again, we have 3*70 = 210.