To determine the number of ways to seat the students, let’s break down the problem step by step.
1. Total Seats and Rows:
- The bus has 25 seats distributed over 5 rows (5 seats per row).
2. Students:
- There are 15 first-grade students and 5 second-grade students.
3. Condition:
- All 5 second-grade students must occupy the first row, which has exactly 5 seats.
4. Seating the Second-Grade Students:
- The number of ways to arrange the 5 second-grade students in the first row’s 5 seats is given by the permutation [tex]\(P(5, 5)\)[/tex].
- This represents the number of ways to order 5 students in 5 seats.
5. Seating the First-Grade Students:
- After placing the second-grade students, there are 20 seats left for the 15 first-grade students.
- The number of ways to arrange 15 first-grade students in the remaining 20 seats is given by the permutation [tex]\(P(20, 15)\)[/tex].
6. Total Number of Seating Arrangements:
- To find the total number of ways to seat all students, multiply the number of ways to seat the second-grade students by the number of ways to seat the first-grade students.
- This gives [tex]\(P(5, 5) \times P(20, 15)\)[/tex].
Based on this step-by-step analysis, the correct answer to the question is:
[tex]\[ B. { }_5 P _5 \times{ }_{20} P _{15} \][/tex]
This choice correctly represents the required permutations to seat the second-grade students in the first row and the first-grade students in the remaining seats.
