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2003 AMC 10A Problem 21

Problem 21 of 25HarderCounting & Probability

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?

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Solution

An assortment is determined by how many of each type are chosen, so we count nonnegative integer solutions to a+b+c=6.a + b + c = 6. By stars and bars, placing 22 dividers among 88 slots gives (82)=28\dbinom{8}{2} = 28 assortments. Thus, the correct answer is D.

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Concepts: stars and bars

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.