Question

In how many different ways can the six faces of cube be painted by 6 different colours? ___

Answer 1

Mark a particular face X. (Later on, we will have to divide whatever answer we get by 6 because all 6 faces are identical at this point, and we have chosen one out of 6 randomly). Put a color on face X. There are 6 ways to do this.
Put another color on the face opposite X. There are 5 ways to do this.

Arrange the remaining 4 colors in the remaining four places - there are 3! ways to do this (circular permutation of 4 colours = (4-1)!
Total $=\frac{[\mathrm{6.5.3}!}{6}$
= 30