2003 AMC 10A Problem 21

Question

Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?

22

25

27

28

729

Answer: D

Concepts:stars and bars

Difficulty rating: 1600

Small hint:

Count the nonnegative integer solutions to

a + b + c = 6

Big hint:

By stars and bars this is

\dbinom{6 + 2}{2}

Solution:

An assortment is determined by how many of each type are chosen, so we count nonnegative integer solutions to

a + b + c = 6.

By stars and bars, placing

2

dividers among

8

slots gives

\dbinom{8}{2} = 28

assortments.

Thus, the correct answer is D.