Assume 11 males audition, one of them being Winston, 4 females audition, one of them being Jackie, and 5 children audition. The casting director has 3 male roles available, 1 female role available, and 2 child roles available.
(1)How many different ways can these roles be filled if exactly one of Winston and Jackie gets a part?
(2)What is the probability (if the roles are filled at random) of both Winston and Jackie getting a part?
To solve both of these questions, we can use the concept of combinations and permutations.
(1) How many different ways can these roles be filled if exactly one of Winston and Jackie gets a part?
To calculate the number of ways the roles can be filled, we need to consider the cases when Winston gets a part and when Jackie gets a part separately:
Case 1: Winston gets a part:
In this case, we have 10 remaining male candidates (excluding Winston), 4 female candidates, and 5 child candidates. We need to fill 2 male roles, 1 female role, and 2 child roles. To calculate the number of ways these roles can be filled, we multiply the number of choices for each role:
Number of ways = (Number of choices for male roles) * (Number of choices for female role) * (Number of choices for child roles)
= C(10,2) * C(4,1) * C(5,2)
= 45 * 4 * 10
= 1800
Case 2: Jackie gets a part:
Similarly, in this case, we have 11 male candidates, 3 remaining female candidates (excluding Jackie), and 5 child candidates. We need to fill 3 male roles, 1 female role, and 2 child roles:
Number of ways = (Number of choices for male roles) * (Number of choices for female role) * (Number of choices for child roles)
= C(11,3) * C(3,1) * C(5,2)
= 165 * 3 * 10
= 4950
To find the total number of ways when exactly one of Winston and Jackie gets a part, we sum up the results from both cases:
Total number of ways = Number of ways (Winston gets a part) + Number of ways (Jackie gets a part)
= 1800 + 4950
= 6750
Therefore, there are 6750 different ways to fill the roles if exactly one of Winston and Jackie gets a part.
(2) What is the probability (if the roles are filled at random) of both Winston and Jackie getting a part?
To calculate the probability, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes = Total number of ways to fill the roles = 6750
Favorable outcomes:
In this case, both Winston and Jackie need to get a part. From our calculation in case 1, we found that there are 1800 ways for Winston to get a part. Similarly, from our calculation in case 2, we found that there are 4950 ways for Jackie to get a part.
Number of favorable outcomes = Number of ways (Winston gets a part) + Number of ways (Jackie gets a part)
= 1800 + 4950
= 6750
Probability (Winston and Jackie both get a part) = Number of favorable outcomes / Total number of possible outcomes
= 6750 / 6750
= 1
Therefore, the probability of both Winston and Jackie getting a part, if the roles are filled at random, is 1 or 100%.