Problem:
Three friends have a total of identical pencils, and each one has at least one pencil. In how many ways can this happen?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The largest number of pencils that any friend can have is four. There are ways that this can happen: , and . There are ways one person can have pencils: , , , , and . There is only one way all three can have two pencils each: . The total number of possibilities is .
The possible distributions of pencils among friends are the following:
The number of possible distributions is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions