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Think about: If the cube is held in a particular orientation, there are 6! ways to paint the six faces. However, if you rotate the cube around, some of these colorings are equivalent. How many colorings are in each equivalence class?
So from what I understand, two colorings are equivalent provided that if the cube is rotated, they have the same coloring.
How can I tackle this problem? I know the strategy for anagrams, but maybe the context of this one is goofing me up. How can I find how many colorings are in each equivalence class?