In how many ways can 6 faces of a rectangular parallelopiped with all 3 dimensions distinct , be painted with 6 different colours??
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Total number of ways by which 6 different colors can paint a cube = 6!
Let the cube be placed in a 3D space.
Consider one of the face that has red color, thus there will be five faces remaining.
The face that is exactly opposite to the red colour face can be coloured in five ways. Thus,
The remaining four faces form a circular permutation, that can be colored in ways = (4-1)!
= 3 !
Therefore, total ways = 5 × 3!
= 30
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