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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. [tex][tex]${ }_{25} P _{20}$[/tex][/tex]
B. [tex][tex]${ }_5 P _5 \times{ }_{20} P _{15}$[/tex][/tex]
C. [tex][tex]${ }_{15} C _{15} \times{ }_{10} C _5$[/tex][/tex]
D. [tex][tex]${ }_{15} P _{15} \times{ }_{10} P _5$[/tex][/tex]
E. [tex][tex]$\quad{ }_{15} P _{15} \times{ }_{10} C _5$[/tex][/tex]

Sagot :

To determine how many ways the students can be seated given the conditions, let's break down the problem into steps:

1. First-Grade Students Placement:
- We need to place 15 first-grade students in the first 3 rows of the bus.
- Each row has 5 seats, giving us [tex]\(3 \text{ rows} \times 5 \text{ seats} = 15 \text{ seats}\)[/tex] reserved for the first-grade students.
- We need to determine how many ways we can arrange 15 students in 15 seats. This is a permutation problem where we are arranging all 15 students in specific positions, which can be computed as [tex]\( {}_{15}P_{15} \)[/tex].

2. Second-Grade Students Placement:
- Next, we have 5 second-grade students who need to be seated in the remaining 2 rows, which provide a total of [tex]\(2 \text{ rows} \times 5 \text{ seats} = 10 \text{ seats}\)[/tex].
- We need to calculate how many ways we can choose 5 seats out of these 10 for the second-grade students and then arrange them. This is computed as [tex]\( {}_{10}P_{5} \)[/tex], representing the number of ways to arrange 5 students out of the 10 remaining seats.

3. Total Number of Ways:
- The total number of ways to seat both the first and second-grade students is the product of the number of ways to arrange the first-grade students and the second-grade students.

The correct mathematical representation of the solution is:
[tex]\[ {}_{15}P_{15} \times {}_{10}P_{5} \][/tex]

Thus, the correct answer corresponds to option:

D. [tex]\({}_{15}P_{15} \times {}_{10}P_{5}\)[/tex]