Problem:
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?
Answer Choices:
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B.
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Solution:
The numbers of the three types of cookies must have a sum of six. Possible sets of whole numbers whose sum is six are
Every ordering of each of these sets determines a different assortment of cookies. There are orders for each of the sets
There are orders for each of the sets
There is only one order for . Therefore the total number of assortments of six cookies is .
Construct eight slots, six to place the cookies in and two to divide the cookies by type. Let the number of chocolate chip cookies be the number of slots to the left of the first divider, the number of oatmeal cookies be the number of slots between the two dividers, and the number of peanut butter cookies be the number of slots to the right of the second divider. For example, represents three chocolate chip cookies, two oatmeal cookies, and one peanut butter cookie. There are ways to place the two dividers, so there are ways to select the six cookies.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions