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]
