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Suppose you are given a cube and 6 different colours. You are asked to colour each face of the cube with a different colour. If so, how many different colorings are possible?

The thing is, this question looks easy to me at first look; the answer seems to be 6! colorings (6 colors for the 1st face, 5 for the next and so on). But the fact that this question appeared in one of the toughest Olympiad level examinations, has made me think otherwise. Can anyone tell me how to solve it?