7. In how many different ways can 7 different prizes be awarded to 5 students in such a way that each student has at least one prize?
7. Ans: 16800
Either one student receives three prizes; or two students are each awarded two prizes. Thus the answer is
$$
\binom{7}{3} \times 5!+\binom{7}{2} \times\binom{ 5}{2} \times \frac{1}{2} \times 5!=16800 .
$$