Fifteen students are going hiking on their spring break. they plan to travel in three vehicles—one seating 7, one seating 5, and one seating 3 students. in how many ways can the students group themselves for their trip?

The students can group themselves in 360360 ways

Step-by-step explanation:

For this exercise we need to use the following equation:

This equation give us the number of assignation of n elements in k cell, where n1, n2, ..nk are the element that are in every cell

In this case we have 15 student that need to be assign in three vehicles with an specific capacity. This vehicles would be the equivalent to cells, so we can write the equation as:

Because the first vehicle have 7 seating, the second vehicle have 5 seating and the third vehicle have 3 seating.

Solving the equation we get 360360 ways to organized 15 students in three vehicles with capacity of 7, 5  and 3 seating.