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A group of seven students is to be seated in a row of seven desks. In how many different ways can the group be seated if two of the preselected students must sit in an end seat (i.e., two students have been preselected to sit in either the first or the seventh seat)?

Correct Answer:

**D**Since the order in which the students are seated matters, this is a permutations problem.

- Seating students in the order A, B, C, D, E, F, G is not the same as seating them B, C, D, E, F, G, A. Since the order in which the students are seated does matter, this is a permutations problem. To solve such a problem, determine the number of possibilities for each slot and then multiply these numbers together to get the total number of permutations.
- There are seven empty seats and seven students to fill them. However, you are told that two of the students must sit in an end seat, either the first seat or the seventh seat. That means that there are 2 possibilities for the first seat – either of these two students. Therefore there is only 2 – 1 = 1 possibility for the seventh seat (i.e., the student who must sit in an end seat who is not sitting in the first seat).
- Since two of the students must sit in end seats, there are 7 – 2 = 5 possibilities for the second seat, 7 – 3 = 4 possibilities for the third seat, 7 – 4 = 3 possibilities for the fourth seat, 7 – 5 = 2 possibilities for the fifth seat, and 7 – 6 = 1 possibility for the sixth seat.
- Now that you have determined the number of possibilities for each slot, multiply them together to determine the total number of permutations:

(2)(5)(4)(3)(2)(1)(1) = 240

So the correct answer is choice (D).