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The edges of a regular pentagon are colored red, blue, or green at random, so that each edge has an equally likely chance of being painted with any given color. What is the probability that in the resulting coloring, no two adjacent edges have the same color?

Aug 3, 2023

#1
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It is given that the edges of a regular pentagon are colored red, blue or green at random, so that each edge has an equally likely chance of being painted with any color.

So you can't have 0 reds.

You can have 1 red and there are 2 scenarios for that.

You can have 2 reds and there are 8 scenarios for that.

You can't have 3+ reds.

In total there are 10 scenarios that work.

Therefore there are 3⁵ - 3 total cases. i.e there are 240 total cases.

So the probability that no two adjacent edges have the same color is,

total scenarios/total cases = 10/240 = 1/24.

Therefore the probability that no two adjacent edges have the same color is 1/24.

Aug 4, 2023
#2
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Can you explain how you found the total number of cases?

Guest Aug 4, 2023