Each off the faces of the cube is coloured by a different colour .How many of the colouringss are distinct?

A cube can be rotated into 6×4=24 configurations (i.e. the red face can be any one of the 6, and then there are 4 ways to rotate it that keep that face red), so the number of different colourings (counting rotations, but not mirror reflections, as the same) is 6!/24=30.



Case n=6: Colour one side with the ugliest colour, and put the cube on a table ugly side down. There are 5 choices for the colour on top. For each of these choices, colour the side facing you with the nicest remaining colour. The last three sides can be coloured in 3! ways, so the number of colourings is (5)(3!).

Case n>6: First choose the colours, then use them. The number of colourings is(n6)(5)(3!).