Six colored cube puzzle


Given six distinct colors, in how many unique ways can a six-faced cube be painted such that no two faces have the same color?

Note: The mixing of colors is not allowed.

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    Suppose we have the colors 1 thru 6.
    Each color is on exactly one face.

    To determine the number of possibilities we rotate the cube so that color 1 is on the bottom face.
    The top face can be any of the remaining 5 colors.
    We know rotate the cube so that color 2 is on the left face (and color 1 still at the bottom).
    If color 2 is at the top face we rotate so that color 3 is the left face.
    Rotating this way doesn’t change the number of possibilities.

    For the front, right and back face we have 3 colors left. And there are 6 ways they can be arranged (3x2x1).

    The total number of unique ways to paint the cube is 30 (5×6).

    CugelTheWise Expert Answered on 25th October 2019.
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