Assume that there are 6 unfilled roles: 1 male and 5 female. There are 5 men and 7 women, including Ann, auditioning for a part in the play.
How many of these casts include Ann?
If Ann is in, then we need 1 male from the 5 men, and 4 more females from the remaining 6 women
Number of casts = 1 x C(5,1) x C(6,4) = 75
or
number of ways with no restriction = C(5,1) x C(7,5) = 105
number of ways without Ann = C(5,1) x C(6,5) = 30
So number of ways with Ann = 105-30 = 75
To determine how many of these casts include Ann, we can use combinatorics.
First, let's consider the number of ways to select a cast without Ann. Since there are 5 men and 7 women, we have a total of 5+7=12 people to choose from.
From these 12 people, we need to select 6 to fill the roles, where only 1 position is for a male and the remaining 5 positions are for females.
The number of ways to select a cast without Ann can be calculated using the combination formula:
C(12,6) = 12! / (6! * (12-6)!)
Expanding this equation:
= 12! / (6! * 6!)
= (12 * 11 * 10 * 9 * 8 * 7 * 6!) / (6! * 6)
= 12 * 11 * 10 * 9 * 8 * 7
= 66,528
Next, let's consider the number of ways to select a cast with Ann. Since there are 5 men and 7 women, to include Ann in the cast, we need to select 1 man and 4 women.
The number of ways to select a cast with Ann can be calculated using the combination formula:
C(5,1) * C(7,4) = (5! / (1! * (5-1)!) * 7! / (4! * (7-4)!))
Expanding this equation:
= (5 * 4 * 3 * 2 * 1 * 7 * 6 * 5) / (4 * 3 * 2 * 1 * 3 * 2 * 1)
= 5 * 7 * 5
= 175
Therefore, the number of casts that include Ann is 175.
To determine how many casts include Ann, we need to consider the different combinations of people that can be selected for the roles, while ensuring that Ann is included.
There are 5 men (excluding Ann) who are auditioning for the part.
Let's break down the possibilities:
1. Ann is chosen as the male role:
Since there is only 1 male role available, Ann can only be chosen as the male role if she is the only one auditioning for that role. Therefore, there is only 1 possible cast in this case.
2. Ann is chosen as one of the 5 female roles:
Since there are 5 female roles available, Ann can be chosen as one of them. In this case, we need to choose 4 out of the remaining 6 women (excluding Ann). This can be done in "6 choose 4" ways, which can be calculated using the combination formula:
C(6, 4) = 6! / (4! * (6-4)!) = 15
Therefore, there are 15 possible casts in this case.
To find the total number of casts that include Ann, we add up the number of possible casts from both cases:
Total = 1 + 15 = 16
Therefore, there are 16 possible casts that include Ann.