I am trying to work this problem out. I know that the systematic counting principle is used. However, I cannot get the right answer.
For the first part, I took
R=2
N=6
6!/(6-2)! (Permutation).
Answer: 30 choices for the Chair/Vice
The second part, I took
R=2
N=8
Combination form.
From here, I do not know what to do or how to get the correct answer. Please explain how to get the answer. Thank You!
The academic computing committee at a college is in the process of evaluating different computer systems. The committee consists of six administrators, six faculty, and two students. A six-person subcommittee is to be formed. The subcommittee must have a chair and vice chair from the administrators, the other four committee members have no particularly defined roles from the faculty and students. In how many ways can this subcommittee be formed?
To solve this problem, we need to use the concept of permutations and combinations.
First, let's find the number of ways to select the chair and vice chair from the administrators.
You correctly used the permutation formula to calculate this:
R = 2 (since we need to select a chair and a vice chair)
N = 6 (since there are 6 administrators)
The formula for permutations is:
P(N, R) = N! / (N - R)!
Plugging in the values, we get:
P(6, 2) = 6! / (6 - 2)!
= 6! / 4!
= (6 x 5 x 4!) / 4!
= (6 x 5)
= 30
So, there are 30 ways to select the chair and vice chair from the administrators.
Now, let's move on to selecting the other four committee members from the combined pool of administrators, faculty, and students.
Since there are 6 faculty members and 2 students, we need to find the number of ways to select 4 members from a pool of 8 (6 faculty + 2 students).
To find this, we can use the combination formula:
C(N, R) = N! / (R!(N - R)!)
Plugging in the values:
C(8, 4) = 8! / (4! (8 - 4)!)
= 8! / (4! x 4!)
= (8 x 7 x 6 x 5 x 4!) / (4! x 4!)
= (8 x 7 x 6 x 5) / (4 x 3 x 2 x 1)
= 70
Therefore, there are 70 ways to select the other four committee members.
To find the total number of ways to form the subcommittee, we multiply the two results together:
Total ways = Number of ways to select chair and vice chair × Number of ways to select other four members
Total ways = 30 × 70
= 2100
Hence, there are 2100 ways to form the subcommittee.