Question

aops several professors attend a committee meeting to review the performance of a special program. each shakes hand once with every other member. after the meeting, three professors are assigned to form a sub-committee to draft the evaluation letter. if there were 15 handshakes in total, in how many ways can this sub-committee be formed? group of answer choices 10

Answer 1

There were 6 professors in total, and there are 20 ways to form a sub-committee of 3 professors from these 6.

Step-by-step explanation:

To determine how many professors there were originally, we can use the formula for the total number of handshakes in a group, which is ½n(n-1), where n is the number of people in the group. Since there were 15 handshakes in total, we can set up the equation ½n(n-1) = 15.

By solving this equation, we find that n(n-1) = 30, which means n² - n - 30 = 0. Factoring this quadratic equation, we get (n-6)(n+5) = 0. The positive solution is n = 6, meaning there were 6 professors in total.

To form a sub-committee of 3 professors from these 6, we can use the combination formula C(n, k) = n! / (k! * (n-k)!), where n is the total number of items to choose from, k is the number of items to choose, and ! denotes the factorial of a number.

Therefore, the number of ways to form the sub-committee is C(6, 3) = 6! / (3! * (6-3)!) = (6 * 5 * 4) / (3 * 2 * 1) = 20. Thus, there are 20 ways to form the sub-committee.