A play contains 15 speaking roles, 10 of them female and 5 male. If the director can choose among 16 female actors and 9 male actors to fill the roles, how many different casts are possible? (NOTE: A cast includes BOTH the actor AND the role that they play.)
I'm just confused on how to set this up. It deals with permutations cause I know there's order, but I'm still not sure what I should be choosing from. I know that I have to multiple the female roles and male roles to get the total, but could you help me set it up? Thank you.
The number of different casts possible is the number of permutations of 16 female actors and 9 male actors for the 15 speaking roles. This can be calculated using the formula:
P(16, 10) x P(9, 5) = 16!/(16-10)! x 9!/(9-5)! = 16!/(6! x 9!/(4! = 16!/(6! x 4!) = 16!/(24 x 4!) = 16!/(96) = 16,796,160
To set up this question, you need to consider a couple of things:
1. Selecting actors for female roles: Since there are 10 female speaking roles and 16 female actors to choose from, you need to find the number of ways to choose 10 actors from a group of 16. This is a combination problem, denoted as "16 C 10" or written as 16 choose 10.
2. Selecting actors for male roles: Similarly, since there are 5 male speaking roles and 9 male actors to choose from, you need to find the number of ways to choose 5 actors from a group of 9. This is also a combination problem, denoted as "9 C 5" or written as 9 choose 5.
To find the total number of different casts, you need to multiply these two combinations:
Total number of casts = (16 C 10) * (9 C 5)
Now, let's calculate these combinations step-by-step:
1. Calculate (16 C 10):
To find (16 C 10), you can use the formula:
(16 C 10) = 16! / (10! * (16 - 10)!)
Where "!" denotes the factorial function.
Calculating the numerator:
16! = 16 * 15 * 14 * 13 * 12 * 11 * 10!
Calculating the denominator:
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Simplifying:
(16 C 10) = (16 * 15 * 14 * 13 * 12 * 11 * 10!) / (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
Canceling out the common terms:
(16 C 10) = (16 * 15 * 14 * 13 * 12 * 11) / (5 * 4 * 3 * 2 * 1)
Calculating the values:
(16 C 10) = 32,432
2. Calculate (9 C 5):
To find (9 C 5), you can use the same formula:
(9 C 5) = 9! / (5! * (9 - 5)!)
Calculating the numerator:
9! = 9 * 8 * 7 * 6 * 5!
Calculating the denominator:
5! = 5 * 4 * 3 * 2 * 1
Simplifying:
(9 C 5) = (9 * 8 * 7 * 6 * 5!) / (5 * 4 * 3 * 2 * 1)
Canceling out the common terms:
(9 C 5) = (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1)
Calculating the values:
(9 C 5) = 1,512
Now, you can multiply these two combinations to find the total number of different casts:
Total number of casts = (16 C 10) * (9 C 5) = 32,432 * 1,512 = 49,100,544
Therefore, there are 49,100,544 different casts possible for this play.
To set up the problem, we can think of it as selecting actors for each role separately. We start by choosing a female actor for each of the 10 female speaking roles. We have 16 female actors to choose from, so for each role, we have 16 choices.
Next, we choose a male actor for each of the 5 male speaking roles. We have 9 male actors to choose from, so for each role, we have 9 choices.
To get the total number of possible casts, we multiply the number of choices for each role together. Since we are choosing independently for the female roles and the male roles, we multiply the number of choices for the female roles by the number of choices for the male roles.
Number of choices for female roles = 16 choices for each of the 10 roles = 16^10.
Number of choices for male roles = 9 choices for each of the 5 roles = 9^5.
Therefore, the total number of possible casts is (16^10) * (9^5).