Question

How many ways can a teacher give 5 different prizes to 5 different of her 25 students

Answer 1

first let's choose the 5 students
number of different groups of 5 are
C(25,5) = 53130
For each of those groups, the five prizes can be given out in 5! ways, (now the order matters)

so final number of ways for the prize - giving is
120(53130) or 6375600 .

Answer 2

120