Assume that there are 6 unfilled roles: 1 male and 5 female. There are 5 men and 7 women, including Ann, auditioning for a part in the play.

Question

How many of these casts include Ann?

Answer 1

If Ann is in, then we need 1 male from the 5 men, and 4 more females from the remaining 6 women

Number of casts = 1 x C(5,1) x C(6,4) = 75

or

number of ways with no restriction = C(5,1) x C(7,5) = 105
number of ways without Ann = C(5,1) x C(6,5) = 30

So number of ways with Ann = 105-30 = 75