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Each face of a cube is painted randomly one of the colors red, orange, yellow, green, blue or purple. What is the probability that the cube has at least one pair of faces that share an edge and are the same color? Express your answer as a decimal to the nearest thousandth.

 Feb 19, 2024
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There are \(6^6\) total ways to color the cube. There are 12 ways to choose 2 adjecent sides in a cube(\({6\choose 3}-3\)) and 6 ways to color them. There are also \(6^4 \) ways to color the rest of the sides but we must take into account symmetry of a cube. There are 24 possible ways to orient a certain coloring of a cube(\(6\cdot4\) 6 ways to choose the base and 4 ways to choose a side) So the total number of ways to get a coloring with atleast one adjacent side is:
\(12\times6\times6^4\div24=\frac{6^5}{2}\)
Therefore:
\(\Large P=\frac{\frac{6^5}{2}}{6^6}=\boxed{\frac{1}{12}}\) Im pretty sure its right

 Feb 19, 2024

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