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.
Out of 16 actors, there are 10 female roles.
Out of 9 actors, there are 5 male roles.
Therefore, 16P10 * 9P5 = Answer
Well, it seems like you're casting quite a theatrical production here! Let's break it down and figure it out together.
To determine the number of different casts possible, we need to consider both the male and female roles and actors separately. So let's start with the female roles:
There are 15 speaking roles in total, and 10 of them are female. Therefore, we need to select 10 female actors from a pool of 16 female actors. This can be represented as a combination (nCr) where n is the total number of actors and r is the number of actors we want to select.
Using the combination formula, we have 16C10, which can be calculated as:
16! / (10! * (16 - 10)!)
Next, let's move on to the male roles:
There are 15 speaking roles in total, and 5 of them are male. Therefore, we need to select 5 male actors from a pool of 9 male actors. Similar to before, this can be represented as a combination, 9C5, which can be calculated as:
9! / (5! * (9 - 5)!)
Finally, to find the total number of different casts possible, we multiply the number of female casts by the number of male casts:
Total casts = 16C10 * 9C5
Calculating these combinations will give you the total number of different casts that are possible for your play. Remember, this assumes that each actor can play only one role. Break a leg with your production!
To determine the number of different casts possible, we need to consider both the number of ways to select the actors/actresses for the roles and the number of ways to assign the roles to them. Let's break it down step-by-step:
Step 1: Determine the number of ways to select female actors for the female roles.
Since there are 10 female roles and 16 female actors, we need to find the number of ways to select 10 actresses from a group of 16. This can be calculated using combinations, denoted as C(n, r), where n is the total number of actors and r is the number of roles to be filled.
In this case, we have C(16, 10) ways to choose the female actors.
Step 2: Determine the number of ways to select male actors for the male roles.
Since there are 5 male roles and 9 male actors, we need to find the number of ways to select 5 actors from a group of 9. Again, this can be calculated using combinations.
In this case, we have C(9, 5) ways to choose the male actors.
Step 3: Multiply the results from Step 1 and Step 2.
To find the total number of different casts, we need to multiply the number of ways to select female actors (Step 1) by the number of ways to select male actors (Step 2).
So the total number of different casts is C(16, 10) * C(9, 5).
Now, let's calculate the values for Step 1 and Step 2:
Step 1: C(16, 10) = 16! / (10!(16-10)!) = 16! / (10! * 6!) = 32,767,680 / (3,628,800 * 720) = 8008
Step 2: C(9, 5) = 9! / (5!(9-5)!) = 15,120 / (120 * 24) = 63
Step 3: Multiply the results from Step 1 and Step 2:
Total number of different casts = 8008 * 63 = 504,504.
Therefore, there are 504,504 different casts possible.
To find the total number of different casts possible, we need to consider the number of ways to select actors for the female speaking roles and the male speaking roles. Since each cast includes both the actor and the role they play, we need to multiply the number of choices for each gender.
Let's start with the female speaking roles. The director can choose among 16 female actors for each role. However, there are 10 female speaking roles to be filled. This means we need to calculate the number of ways to select 10 actresses from a group of 16. This is known as a combination because the order of selection doesn't matter. We can calculate this using the formula for combinations:
C(n, r) = n! / (r!(n-r)!)
Here, n represents the total number of options available (16 female actors), and r represents the number of choices we need to make (10 female speaking roles). Plugging these values into the formula, we get:
C(16, 10) = 16! / (10!(16-10)!) = 16! / (10!6!)
Similarly, for the male speaking roles, the director can choose among 9 male actors for each role, and there are 5 male speaking roles to be filled. Using the same formula, we have:
C(9, 5) = 9! / (5!(9-5)!) = 9! / (5!4!)
To find the total number of different casts possible, we multiply the number of choices for the female speaking roles by the number of choices for the male speaking roles:
Casts = C(16, 10) * C(9, 5)
Substituting the values we obtained earlier:
Casts = (16! / (10!6!)) * (9! / (5!4!))
You can evaluate this expression using a calculator to find the total number of different casts possible.