(1) How many different casts are there? (2) How many of these casts include Sally?

Question

involving the casting of a play in a community theater. Assume that there are 8 unfilled roles: 3 male and 5 female. There are 6 men and 9 women, including Sally, auditioning for a part in the play.

Answer 1

You have to figure the number of way to assign the male roles and female roles separately and then multiply to find the total number of casts.

 

So, with 6 men and 3 male roles, there are 6! / (3! (6-3)!) = 6! / (3!3!) = 20 different ways to assign the male roles.

 

Similarly there are 9! / (5! (9-5!)) = 9! / (5!4!) = 126 different ways to assign the female roles.

 

So there are 20*126 = 2520 different casts.

 

To find how many casts include Sally, assume that she has one of the roles and then find the number of ways to assign the other 4 female roles to the other 8 women.  That's 8! / ((4!) (8-4)!) = 8! / (4!4!) = 70.  So multiply that by the number of ways to assign the male roles (20) to get 1400 different casts that include Sally.