M01-14

Official Solution:

Rachel wants to divide 21 beans into three piles of seven beans each. She has 10 red beans, 5 yellow beans, and 6 green beans. If each pile must contain at least one bean of each color, what is the maximum number of red beans that can she put in one of the piles?

A. 5
B. 6
C. 8
D. 9
E. 10


To maximize the number of red beans in a single pile, it is necessary to minimize the number of other colored beans in that same pile. As each pile must contain at least one bean of every color, the minimum number of yellow and green beans in a pile is one each. Given the rule that each pile must contain exactly 7 beans, the highest possible number of red beans in one pile is, therefore, 7 - 1 - 1 = 5. This can be accomplished by having one pile with 1 yellow, 1 green, and 5 red beans, while still distributing the remaining 14 beans (5 red, 4 yellow, and 5 green) across the other two piles so that each of them also includes at least one bean of each color.


Answer: A