Problem:
Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
There are two cases to consider.
: One student receives award and the other two students receive awards each. There are ways to select the student who will receive award and choices for that student's award. That leaves awards to distribute evenly. For the next student there are ways to choose that student's awards, leaving the remaining awards for the third student. Therefore the number of ways to give award to one student and awards to each of the other two students is .
: One student receives awards and the other two students receive award each. There are ways to select the student who receives awards and ways to choose that student's awards. The remaining awards can be distributed to the other two students in ways. Therefore the number of ways to give awards to one student and award to each of the other two students is .
In total there are ways to distribute the awards.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions