Answer:
To determine the number of ways the students can be seated if all the first-grade students occupy the first 3 rows of the school bus, we can break down the problem into two steps:
Step 1: Arrange the first-grade students in the first 3 rows
Since there are 15 first-grade students and 15 seats available in the first 3 rows, we can use the permutation formula to calculate the number of ways they can be arranged. The formula for permutations is nPr = n! / (n - r)!, where n is the total number of items and r is the number of items to be arranged.
Using the permutation formula, we have:
15P15 = 15!
= 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
≈ 1.30767 x 10^12
Step 2: Arrange the second-grade students in the remaining seats
After arranging the first-grade students, we are left with 10 seats for the 5 second-grade students. We can use the combination formula to calculate the number of ways they can be arranged. The formula for combinations is nCr = n! / (r!(n - r)!), where n is the total number of items and r is the number of items to be chosen.
Using the combination formula, we have:
10C5 = 10! / (5!(10 - 5)!)
= 10! / (5! x 5!)
= (10 x 9 x 8 x 7 x 6) / (5 x 4 x 3 x 2 x 1)
= 252
Therefore, the total number of ways the students can be seated is the product of the arrangements of the first-grade students and the combinations of the second-grade students:
1.30767 x 10^12 x 252 ≈ 3.29725 x 10^14
Based on the given answer choices, the closest option to this value is:
D. 15P15 * 10P5
Explanation: