Question

Three students (S) and six faculty members (F) are on a panel discussing a new college policy (a) In how many different ways can the nine participants be lined up at a table in the front of the auditorium? (b) How many lineups are possible, considering only the labels S and F? (c) For each of the nine participants, you are to decide whether the participant did a good job or a poor job stating his or her opinion of the new policy; that is, give each of the nine participants a grade of G or P. How many different scorecards are possible?

Answer 1

The number of ways to line up the participants at the table in the front of the auditorium, considering different scenarios and labels, and the total number of different scorecards that can be created for the participants.

Step-by-step explanation:

(a) The number of ways to line up the nine participants is calculated using the formula for permutations. Since the order matters, we use the permutation formula: P(n, r) = n! / (n - r)!. In this case, n = 9 and r = 9, so the number of lineups is 9! = 362,880.

(b) Considering only the labels S and F, we have 3 students and 6 faculty members. The number of ways to line up the participants is calculated using the permutation formula: P(n, r) = n! / (n - r)!. In this case, n = 9 and r = 3, so the number of lineups is 9! / 3! = 504.

(c) For each of the nine participants, there are two possible grades: G (good job) or P (poor job). Therefore, the number of different scorecards is 2^9 = 512.