7.A group has 9 women and 7 men.
d. Suppose 2 members of the group refuse to work together. How many subgroups of 5 can be chosen
8. In how many ways can 16 people be seated:
a. In a row, if 4 of the 16 do not want to sit next to one another
b. In a row, if 3 of the 16 must be seated next to one another
c. In a circle, if 3 of the 16 must be seated next to one another.
d. In a circle, if 4 of the 16 do not want to sit next to one another.
For question 7:
d. To determine the number of subgroups of 5 that can be chosen from a group of 9 women and 7 men, we can use the concept of combinations.
First, we need to calculate the number of ways to choose 5 people from the total of 16 (9 women + 7 men).
The formula to calculate combinations is given by:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of objects and r is the number of objects to be chosen.
Using this formula, we can calculate the number of subgroups of 5 as:
C(16, 5) = 16! / (5!(16-5)!)
Simplifying this expression:
C(16, 5) = 16! / (5!11!)
Now, we can calculate the value using any suitable method, such as using a calculator, or by simplifying the factorial expressions manually.
For question 8:
a. To determine the number of ways to seat 16 people in a row, with 4 people not sitting next to each other, we can use the concept of permutations with restrictions.
First, let's consider the 4 people who do not want to sit next to each other as a single entity. So, we have 1 group of 4 + 12 people, which can be seated in (13-4)! = 9! ways.
Within the group of 4 people who don't want to sit together, they can be arranged in 4! ways among themselves.
Therefore, the total number of seating arrangements is 9! * 4!.
b. To determine the number of ways to seat 16 people in a row, with 3 people seated next to each other, we can treat the group of 3 people as a single entity. So, we have 1 group of 3 + 13 people, which can be seated in (14-3)! = 11! ways.
Within the group of 3 people who must sit together, they can be arranged in 3! ways among themselves.
Therefore, the total number of seating arrangements is 11! * 3!.
c. To determine the number of ways to seat 16 people in a circle, with 3 people seated next to each other, we can treat the group of 3 people as a single entity. So, we have 1 group of 3 + 13 people, which can be seated in (14-3)! = 11! ways.
Within the group of 3 people who must sit together, they can be arranged in 3! ways among themselves.
Since it is a circle, we need to divide the total number of seating arrangements by the number of rotations. In this case, there are 16 possible rotations for each seating arrangement.
Therefore, the total number of seating arrangements is 11! * 3! / 16.
d. To determine the number of ways to seat 16 people in a circle, with 4 people not sitting next to each other, we can use the concept of permutations with restrictions.
First, let's consider the 4 people who do not want to sit next to each other as a single entity. So, we have 1 group of 4 + 12 people, which can be seated in (13-4)! = 9! ways.
Within the group of 4 people who don't want to sit together, they can be arranged in 4! ways among themselves.
Since it is a circle, we need to divide the total number of seating arrangements by the number of rotations. In this case, there are 16 possible rotations for each seating arrangement.
Therefore, the total number of seating arrangements is 9! * 4! / 16.