Answer
See below
Work Step by Step
We know that if we want to select $r$ objects out of $n$ disregarding the order, we can do this in $_nC_r=\frac{n!}{r!(n-r)!}$ ways.
Choose 5 out of 15 students to wash fire trucks. Here we have $n_1=15,r_1=5$: $_{15}C_5=\frac{15!}{10!5!}=3003$
Choose 7 out of 10 students to repair the station's interior. Here we have $n_1=10,r_1=7$: $_{15}C_5=\frac{10!}{3!7!}=120$
Choose 3 out of 3 students to do the maintenance. Here we have $n_1=3,r_1=3$: $_{15}C_5=\frac{3!}{0!3!}=1$
The number of possible job assignments is $_{15}C_5 \times_{10}C_7\times_3C_3=3003\times120\times1=360360$
