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 can use the concepts of permutations and combinations.
Let's break down the problem step by step:
1. Selecting the chair and vice chair from the administrators:
Since we need to select two people from the six administrators, and the order matters (first chair, second vice chair), we can use the permutation formula. In this case, R = 2 (since we're selecting two people) and N = 6 (total number of administrators).
Using the permutation formula, we get 6P2 = 6! / (6-2)! = 6! / 4! = (6 * 5 * 4 * 3 * 2 * 1) / (4 * 3 * 2 * 1) = 30.
So there are 30 ways to select the chair and vice chair from the administrators.
2. Selecting the other four committee members from the faculty and students:
Since the roles of these committee members are not defined, we don't need to consider the order in which they are selected. Therefore, we can use the combination formula. In this case, R = 4 (since we need to select four people) and N = 6 administrators + 6 faculty + 2 students = 14.
Using the combination formula, we get 14C4 = 14! / (4! * (14-4)!) = (14 * 13 * 12 * 11) / (4 * 3 * 2 * 1) = 1001.
So there are 1001 ways to select the other four committee members from the faculty and students.
Now, to find the total number of ways to form the subcommittee, we multiply the results from the two steps above:
Total number of ways = 30 (chair and vice chair) * 1001 (other committee members) = 30,030.
Therefore, there are 30,030 ways to form the subcommittee.