Painting the faces of a cube with distinct colours

Solution 1:

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 $$\binom{n}{6}(5)(3!).$$

Solution 2:

A cube can be rotated into $6 \times 4 = 24$ 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 $6!/24 = 30$.