Problem:
Each vertex of convex pentagon is to be assigned a color. There are 6 colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
Answer Choices:
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Solution:
If five distinct colors are used, then there are different color choices possible. They may be arranged in ways on the pentagon, resulting in colorings.
If four distinct colors are used, then there is one duplicated color, so there are different color choices possible. The duplicated color must appear on neighboring vertices. There are 5 neighbor choices and ways to color the remaining three vertices, resulting in a total of colorings.
If three distinct colors are used, then there must be two duplicated colors, so there are different color choices possible. The non-duplicated color may appear in 5 locations. As before, a duplicated color must appear on neighboring vertices, so there are 2 ways left to color the remaining vertices. In this case there are colorings possible.
There are no colorings with two or fewer colors. The total number of colorings is .
The problems on this page are the property of the MAA's American Mathematics Competitions