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

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%

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.

Ya well 24192 isn't correct.

The total ways is 18564 man.