Question

a committee consisting of 4 faculty members and 5 students is to be formed. every committee position has the same duties and voting rights. there are 7 faculty members and 13 students eligible to serve on the committee. in how many ways can the committee be formed?

Answer 1

To determine the number of ways to form the committee with 4 faculty members and 5 students, we need to calculate the combinations of faculty members and students separately and then multiply them together.

For faculty members:
There are 7 eligible faculty members, and we need to select 4 of them for the committee. This can be done using the combination formula:

C(7, 4) = 7! / (4!(7 - 4)!) = 35.

So, there are 35 ways to select 4 faculty members from the 7 eligible ones.

For students:
There are 13 eligible students, and we need to select 5 of them for the committee. Using the combination formula again:

C(13, 5) = 13! / (5!(13 - 5)!) = 1287.

Therefore, there are 1287 ways to select 5 students from the 13 eligible ones.

To form the committee, we multiply the number of ways to select faculty members by the number of ways to select students:

35 * 1287 = 44,895.

Hence, there are 44,895 ways to form the committee consisting of 4 faculty members and 5 students from the eligible candidates.

Answer 2

C(7, 4) = 7! / (4!(7-4)!) = (7 * 6 * 5) / (3 * 2 * 1) = 35

and

C(13, 5) = 13! / (5!(13-5)!) = (13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1) = 1287

Multiply these two combinations since the faculty members and students are selected independently:

Total number of ways to form the committee = C(7, 4) * C(13, 5) = 35 * 1287 = 45,045

Therefore, there are 45,045 ways to form the committee consisting of 4 faculty members and 5 students.