Symmetry is a Big Problem
Probability
Level
4
Each of the faces of a cube is colored one of 6 distinct colors. How many of the colorings are distinct?
The answer is 30.
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1 solution
Call the six colors 1,2,3,4,5,6.Put the cube on the table so that face 1 is at the bottom. Consider face 2. If it is at the top then we can rotate the cube about a vertical axis so that face 3 is in front.Now the cube is fixed There are 3!=6 ways to complete the coloring.
Now suppose the face 2 is a neighbour of 1.Then we rotate the cube so that 2 is in front.Now that cube is fixed and the coloring can be completed in 4!=24 ways.
Altogether there are 6+24=30 distinct colorings of the cube by six colors.