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.
I'll tackle 7d.
Out of 7 people, two do not want to work together.
Method 1:
Calculate all possible ways (5 out of 7), and subtract those that include the two particular persons (3 out of 5).
C(7,5)-C(5,3)
=7!/(5!2!)+5!/(3!2!)
=11
Method 2:
Make 3 cases,
1. without A, nor B : 1 way
2. with only A and not B:
C(5,4)=5
3. with only B and not A:
C(5,4)=5
Total: 1+5+5=11
d. Suppose 2 members of the group refuse to work together. How many subgroups of 5 can be chosen?
To solve this problem, we can consider the total number of subgroups without any restrictions and subtract the number of subgroups that contain the 2 members who refuse to work together.
Step 1: Calculate the total number of subgroups of 5 that can be chosen from a group of 16 (9 women + 7 men).
We use the combination formula: C(n, r) = n! / (r!(n-r)!)
C(16, 5) = 16! / (5!(16-5)!)
= 16! / (5! * 11!) = (16 * 15 * 14 * 13 * 12) / (5 * 4 * 3 * 2 * 1) = 4368
There are 4368 subgroups of 5 that can be chosen from the group of 16.
Step 2: Calculate the number of subgroups that contain the two members who refuse to work together.
Since the two members refuse to work together, they must always be selected together. So, we treat them as a single unit and calculate the number of subgroups of 4 that can be chosen from the remaining 14 members.
C(14, 4) = 14! / (4!(14-4)!)
= 14! / (4! * 10!) = (14 * 13 * 12 * 11) / (4 * 3 * 2 * 1) = 1001
There are 1001 subgroups of 4 that contain the two members who refuse to work together.
Step 3: Subtract the number of subgroups that contain the two members who refuse to work together from the total number of subgroups.
Total number of subgroups of 5 without any restrictions = 4368
Number of subgroups that contain the two members who refuse to work together = 1001
Number of subgroups of 5 that can be chosen from the group of 16 with the given restriction = 4368 - 1001
= 3367
Therefore, there are 3367 subgroups of 5 that can be chosen from the group of 16, given that 2 members refuse to work together.
Now, let's move on to question number 8.