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 of the second-grade students occupy the first row?
To calculate the number of ways that the students can be seated, we can use the concept of permutations.
First, let's consider the first-grade students. We have 15 first-grade students who need to be seated in 20 seats (since the first row is already occupied by second-grade students).
We can calculate the number of ways to arrange the first-grade students in these 20 seats by using the permutation formula: P(n, r) = n! / (n - r)!, where n is the total number of items and r is the number of items we want to arrange.
In this case, we want to arrange 15 first-grade students in 20 seats, so the calculation would be P(20, 15) = 20! / (20 - 15)!.
Calculating this formula, we get:
P(20, 15) = 20! / 5!
Next, let's consider the second-grade students. Since all of them occupy the first row, they do not need to be rearranged.
Therefore, the total number of ways the students can be seated is equal to the number of ways to arrange the first-grade students multiplied by 1 (since the second-grade students do not need rearrangement).
So the final calculation would be:
Total ways = P(20, 15) * 1
By evaluating this expression, we can determine the total number of ways the students can be seated.