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:
First, we can calculate the number of ways to just give out awards, without making sure every student has at least one. This is given by .
Next, we subtract the number of distributions where not everyone gets at least one award. This can be counted as the number of distributions with at most two people receiving all the awards. We can sort this by two cases:
Case 1: If two people have all the awards, there are ways to distribute the awards amongst these two people. However, we must subtract as that corresponds to one person owning all the awards, which is outside the definition of this case. This gives us combinations. Finally, we multiply this by , as there are ways to choose the group of two people. This case yields distributions.
Case 2: If one person has all the awards, there is exactly way to distribute the awards amongst that person. Finally, we multiply this by , as there are ways to choose the person. This case yields distributions.
Thus, there are a total of cases to remove.
Therefore, the final number of valid distributions is .
Thus, the correct answer is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions