Five separate awards (best scholarship, best leadership qualities, and so on) are to be presented to selected students from a class of 20. how many different outcomes are possible if (a) a student can receive any number of awards? (b) each student can receive at most 1 award?

Case a: A student can receive any number of awards

Let's count our choices: the first award can go to any of the 20 students. So we have 20 choices. The second awards can also go to any of the 20 students. So we have 20*20 choices for the first two awards. Similarly, we have 20*20*20 choices for the first three awards, and so on.

So, there are possible ways to give the awards, if a student can receive as many awards as possible.

Case b: A student can receive only one awards

This will be very similar to the previous case, but with a minor restriction: as before, we have 20 choices for the first award, because it can go to any of the 20 students.

But when it comes to the second award, we only have 19 choices, because we can't give it to the student who already won the first award.

Similarly, we can give the third award to one of the 18 remaining students, because we can't give it to the students who already won the first or second award.

So, in the end, we have

ways of awarding the students, if a student can win only one award.