To solve this problem, we need to determine the number of ways to seat students from the first grade and the second grade as specified.
1. First-Grade Students:
- There are 15 first-grade students.
- They are to be seated in the first 3 rows, which total 15 seats (since each row has 5 seats and 3 rows contribute 3 * 5 = 15 seats).
- We need to calculate the number of ways to arrange 15 first-grade students in these 15 seats.
- The number of ways to arrange 15 students in 15 seats is given by the permutation [tex]\( \text{P}(15, 15) \)[/tex].
2. Second-Grade Students:
- There are 5 second-grade students.
- They will occupy 5 of the 10 remaining seats (since the total number of seats is 25 and 15 are already occupied by the first-grade students, leaving us 25 - 15 = 10 seats).
- We need to calculate the number of ways to arrange 5 second-grade students in these 10 remaining seats.
- The number of ways to arrange 5 students in 10 seats is given by the permutation [tex]\( \text{P}(10, 5) \)[/tex].
Thus, the total number of ways to seat all the students while satisfying the given conditions is the product of the two permutations:
[tex]\[ \text{P}(15, 15) \times \text{P}(10, 5) \][/tex]
According to the answer choices provided:
- Option D corresponds to [tex]\( \text{P}(15, 15) \times \text{P}(10, 5) \)[/tex].
Therefore, the correct answer is:
D. [tex]\( \text{P}(15, 15) \times \text{P}(10, 5) \)[/tex]
