Question

A school bus has 25 seats, with 5 rows of 5 seats. 15 students from the first grade and 5 students from the second-grade travel in the bus. How many ways can the students be seated if all the first-grade students occupy the first 3 rows?

A. 25P20
B. 5P5 * 20P15
C. 15C15 * 10C5
D. 15P15 * 10P5
E. 15P15 * 10C5

Answer 1

The correct answer option is D. 15P15 x 10P5.

Explanation:

We know that there are 15 students in first grade and we have 5 rows of 5 seats to accommodate them. So first grade students can be arranged to occupy the seats in
`15P15`.

Also, we have 5 students in the second grade with a total of 25 seats from which 15 seats are already occupied so we are left with 10 seats now.

Therefore, the students can be seated in 15P15 x 10P5 ways.

Answer 2

D. 15P15 * 10P5

Explanation:

Since you have to place all first-grade students in the first three rows, and nowhere else, we have to make a special calculation for that, then another for the rest of the bus.

These are permutations since the order is important. If we sit John, Paul, Ringo, George and Pete in this order in the first row it's a different way than seating them (in the same order) in the second row for example.

For the 15 first-graders of the first three rows (15 seats), we have 15P15 since all 15 places have to be occupied by all 15 first-graders.

Then we have 10 remaining seats left to be assigned to the 5 second-graders. That is 10P5.

We then multiply the permutation numbers of those two arrangements to get the total ways:

15P15 * 10P5, answer D.