5 actors and 8 actresses are available for a play which requires 3 male roles and 4 female roles. Find the number of different possible cast lists?
To find the number of different possible cast lists, we need to determine how many ways we can select 3 males from 5 actors and 4 females from 8 actresses.
First, let's calculate the number of ways to select 3 males from 5 actors. This can be done using a combination formula. The number of ways to choose k items from n items is given by the formula:
C(n, k) = n! / (k! * (n - k)!)
In our case, n = 5 (the number of actors) and k = 3 (the number of male roles). Thus, the number of ways to select 3 males from 5 actors is:
C(5, 3) = 5! / (3! * (5 - 3)!) = 5! / (3! * 2!) = (5 * 4 * 3!) / (3! * 2 * 1) = (5 * 4) / (2 * 1) = 10
So, there are 10 ways to select 3 males from 5 actors.
Next, let's determine the number of ways to select 4 females from 8 actresses. Using the same combination formula:
C(n, k) = n! / (k! * (n - k)!)
In this case, n = 8 (the number of actresses) and k = 4 (the number of female roles). Thus, the number of ways to select 4 females from 8 actresses is:
C(8, 4) = 8! / (4! * (8 - 4)!) = 8! / (4! * 4!) = (8 * 7 * 6 * 5 * 4!) / (4! * 4 * 3 * 2 * 1) = (8 * 7 * 6 * 5) / (4 * 3 * 2 * 1) = 70
So, there are 70 ways to select 4 females from 8 actresses.
To determine the number of different possible cast lists, we multiply the number of ways to select males by the number of ways to select females:
Total number of possible cast lists = Number of ways to select 3 males * Number of ways to select 4 females = 10 * 70 = 700
Therefore, there are 700 different possible cast lists.
To find the number of different possible cast lists, we need to calculate the combinations of actors and actresses that can fill the required number of male and female roles.
The number of ways to choose 3 actors out of 5 is given by the combination formula: C(5, 3) = 5! / (3! * (5-3)!) = 10.
Similarly, the number of ways to choose 4 actresses out of 8 is: C(8, 4) = 8! / (4! * (8-4)!) = 70.
To find the total number of possible cast lists, we need to multiply the number of ways to choose actors and actresses.
Total number of possible cast lists = 10 * 70 = 700.
Therefore, there are 700 different possible cast lists for the play.