assume 8 males audition,
one of them being Miles, 6 females audition,
one of them being Jackie, and 4 children audition.
The casting director has 3 male roles available,
1 female role available,
and 2 child roles available.
How many different ways can these roles be filled if exactly one
of Miles and Jackie gets a part?
What is the probability (if the roles are filled at random) of both
Miles and Jackie getting a part?
I know the total ways the parts can be filled is 24192
10 years ago
9 years ago
The total ways is 18564 man.
1 year ago
To find the number of different ways the roles can be filled if exactly one of Miles and Jackie gets a part, we need to consider three cases:
Case 1: Miles gets a part and Jackie doesn't
In this case, the 3 male roles will be filled from the remaining 7 males (excluding Miles), the female role will be filled from the 6 females, and the 2 child roles will be filled from the 4 children. The number of ways this can be done is:
7C3 (for 3 males) * 6C1 (for 1 female) * 4C2 (for 2 children) = 35 * 6 * 6 = 1260
Case 2: Jackie gets a part and Miles doesn't
In this case, the 3 male roles will be filled from the 8 males, excluding Miles, the female role will be filled from the 5 females (excluding Jackie), and the 2 child roles will be filled from the 4 children. The number of ways this can be done is:
7C3 (for 3 males) * 5C1 (for 1 female) * 4C2 (for 2 children) = 35 * 5 * 6 = 1050
Case 3: Both Miles and Jackie get a part
In this case, the 3 male roles will be filled from the remaining 6 males (excluding Miles and Jackie), the female role will be filled from the remaining 6 females (excluding Miles and Jackie), and the 2 child roles will be filled from the 4 children. The number of ways this can be done is:
6C3 (for 3 males) * 6C1 (for 1 female) * 4C2 (for 2 children) = 20 * 6 * 6 = 720
Therefore, the total number of different ways the roles can be filled if exactly one of Miles and Jackie gets a part is:
1260 (from Case 1) + 1050 (from Case 2) + 720 (from Case 3) = 3030
To find the probability of both Miles and Jackie getting a part, we need to consider the last case where both of them get a part. The total number of ways the roles can be filled is still 24192. Therefore, the probability is:
Number of ways both Miles and Jackie get a part / Total number of ways the roles can be filled
= 720 / 24192 = 0.0297 or 2.97%
11 months ago
To find the number of different ways the roles can be filled if exactly one of Miles and Jackie gets a part, we need to consider all possible scenarios where one of them gets a part and the others do not.
First, let's analyze the number of ways the roles can be filled if Miles gets a part:
- There are 7 remaining males (excluding Miles) who can fill the other two male roles. This can be done in C(7, 2) = 21 ways.
- There are 6 females vying for the one female role, so this role can be filled in 6 ways.
- There are still 4 children and 2 child roles available. The 4 children can be assigned to these roles in C(4, 2) = 6 ways.
Now, let's analyze the number of ways the roles can be filled if Jackie gets a part (and Miles doesn't):
- There are still 8 males available, out of which 2 can be selected for the male roles in C(8, 2) = 28 ways.
- There are 5 females left, so the one female role can be filled in 5 ways.
- There are still 4 children, out of which 2 can be chosen for the child roles in C(4, 2) = 6 ways.
Since the number of ways for each scenario is independent, we need to add these two cases together to find the total number of ways: 21 + 6 + 28 + 5 + 6 = 66 ways.
Therefore, the number of different ways the roles can be filled if exactly one of Miles and Jackie gets a part is 66.
To find the probability of both Miles and Jackie getting a part, we need to calculate the probability of one of them getting a part and multiply it by the probability of the other one getting a part.
The probability of Miles getting a part is given by the number of ways he can get a part (66) divided by the total number of ways the roles can be filled (24192). So, the probability of Miles getting a part is 66/24192.
Similarly, the probability of Jackie getting a part is 66/24192.
To find the probability of both Miles and Jackie getting a part, we multiply their probabilities: (66/24192) * (66/24192) = 4356/5832 = 0.7477 (approximately).
Therefore, the probability of both Miles and Jackie getting a part is approximately 0.7477.