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 of the second-grade students occupy the first row?

A. 25P20
B. 5P5 × 20P15
C. 5C5 × 25C14
D. 5P5 × 15P15
E. 5P5 × 25C15

Answer 1

`^5P_5 * ^(20)P_(15)`

Explanation:

No. of students from first grade = 15

No. of students from second grade = 5

There are 5 rows of seats

Each row contains 5 seats

Total seats = 25

No. of ways for second-grade students occupy the first row(i.e first 5 seats) =
`^5P_5`

Remaining seats = 20

No. of students from first grade = 15

So, No. of ways for first-grade students occupy remaining seats =
`^(20)P_(15)`

So, No. of ways for the students can be seated if all of the second-grade students occupy the first row =
`^5P_5 * ^(20)P_(15)`

So, Option B is true

Hence No. of ways for the students can be seated if all of the second-grade students occupy the first row is
`^5P_5 * ^(20)P_(15)`

Answer 2

The answer is B. 5P5 x 20P15.

This is because to make the second grade students to sit in the first row we have 5 seats and 5 students so we will permute the 5 students to those 5 seats. So 5P5.

Now we are left with 20 seats and 15 first grade students so we can simply permute those 15 students into those 20 seats. So 20P15.

Finally using the counting rule principle we will multiply both of these so 5P5 x 20P15.