Okay,so: Assume that there are 8 unfilled roles:
3 male and 5 female. There are 5 men and 6 women, including Sally, auditioning for a part in the play.
How many of these casts include Sally?
since Sally has to be chosen, we need 4 more females from the 5 remaining females, and 3 of the 5 men
number of casts with Sally
= 1*C(5,4) * C(5,3)
= 1*5*10 = 50
To determine the number of casts that include Sally, you need to select a certain number of individuals from the group of men and women who are auditioning, and ensure that Sally is one of those selected.
First, let's calculate the number of different ways you can select a certain number of men and women from the group of auditionees. This can be done using the combinations formula, which is denoted as:
nCr = n! / (r!(n-r)!)
In this case, since you have 5 men auditioning and you need to select a certain number of them, you can use this formula. Similarly, you have 6 women auditioning, and you will need to select a certain number of them too.
Now, there are different possible scenarios depending on the number of men and women you select. Let's calculate the total number of casts that include Sally for each scenario, and then sum them up to get the final answer.
1) If you select no men and only Sally, there is only 1 way to select Sally.
Number of casts = 1
2) If you select 1 man and Sally, you can choose one man from the 5 available and Sally from the 6 women.
Number of casts = 5C1 * 6C0 = 5
3) If you select 2 men and Sally, you can choose two men from the 5 available and Sally from the 6 women.
Number of casts = 5C2 * 6C0 = 10
4) If you select 3 men and Sally, you can choose three men from the 5 available and Sally from the 6 women.
Number of casts = 5C3 * 6C0 = 10
Now, summing up the casts from each scenario:
Total number of casts including Sally = 1 + 5 + 10 + 10 = 26
So, there are 26 different casts that include Sally.
To determine the number of casts that include Sally, we need to consider the number of possible combinations.
We have 5 men and 6 women auditioning for a total of 11 people.
The number of casts that include Sally can be calculated by combining the remaining roles after Sally has been selected.
We have 3 male roles and 5 female roles remaining.
The number of casts that include Sally can be found using the formula: M × W, where M is the number of remaining male roles and W is the number of remaining female roles.
In this case, we have 3 male roles remaining and 5 female roles remaining.
Thus, the number of casts that include Sally can be calculated as:
3 × 5 = 15
Therefore, there are 15 casts that include Sally.