In how many ways can a cube be coloured using 5 distinct colours, if one colour is used on two opposite faces and each of the remaining colours has to be used exactly once?

Since, a cube is symmetrical at each side, there is only 1 way to colour the first face. The opposite face of the cube must be coloured with the same colour, which can be done in only 1 way. Now, the colour which is repeated on the two opposite faces can be chosen in 5 ways. Again, from the remaining four faces, a face can be picked in only 1 way. After colouring three faces of the cube, the remaining faces would appear distinct. So, number of ways in which remaining four faces can be coloured = 4! = 24 But these arrangements are counted 4 times due to rotational symmetry of the cube. So, distinct arrangements  = 24 ÷ 4 = 6 So, required number of ways  = 5 × 6 = 30