Finite, Math Question

Question

Assume that there are 8 unfilled roles: 3 male and 5 female. There are 5 men and 8 women, including Mary, auditioning for a part in the play.

There are 403200 different casts.

How many of these casts include Mary?

Answer 1

To calculate the number of casts that include Mary, we can use combinations. There are two cases to consider:

1. Mary is selected from the 5 men.
2. Mary is selected from the 8 women.

The total number of ways to form a cast without considering Mary is (83)(38​) since there are 3 male roles to be filled from 5 men and 5 female roles to be filled from 8 women.

Now, for each of these casts, we need to consider Mary. There are two sub-cases:

1. Mary is included in the male roles. This can be done in (42)(24​) ways since there are 2 more male roles to be filled from the remaining 4 men (excluding Mary).
2. Mary is included in the female roles. This can be done in (74)(47​) ways since there are 4 more female roles to be filled from the remaining 7 women (excluding Mary).

The total number of casts with Mary is given by:

(83)×((42)+(74))(38​)×((24​)+(47​))

Let's calculate this:

(83)=8!3!×(8−3)!=56(38​)=3!×(8−3)!8!​=56

(42)=4!2!×(4−2)!=6(24​)=2!×(4−2)!4!​=6

(74)=7!4!×(7−4)!=35(47​)=4!×(7−4)!7!​=35

Now, multiply these values:

56×(6+35)=56×41=229656×(6+35)=56×41=2296

Therefore, there are 2296 different casts that include Mary.