Solution:

Assuming that the 6 colours are distinct, we can paint the faces of the cube in the following ways:

1. All faces painted with the same colour: There are 6 ways to choose the colour and only 1 way to paint the cube.

2. 5 faces painted with one colour and 1 face painted with another colour: There are 6 ways to choose the colour for the 5 faces and 5 ways to choose the colour for the remaining face. However, since the cube can be rotated in 6 different ways, we need to divide by 6 to avoid overcounting. Therefore, the total number of ways is (6 x 5) / 6 = 5.

3. 4 faces painted with one colour and 2 faces painted with another colour: There are 6 ways to choose the colour for the 4 faces and 5 ways to choose the colour for the remaining 2 faces. Again, we need to divide by 6 to avoid overcounting. Therefore, the total number of ways is (6 x 5) / 6 x 5 / 6 = 25 / 6.

4. 3 faces painted with one colour and 3 faces painted with another colour: There are 6 ways to choose the colour for the 3 faces and (5 x 4) / 2 ways to choose the colours for the remaining 3 faces. Since the cube can be rotated in 6 different ways, we need to divide by 6 to avoid overcounting. Therefore, the total number of ways is 6 x (5 x 4) / 2 / 6 = 60 / 6 = 10.

5. 2 faces painted with one colour, 2 faces painted with another colour, and 2 faces painted with a third colour: There are (6 x 5) / 2 ways to choose the colours for the 3 pairs of faces. However, since the cube can be rotated in 6 different ways, we need to divide by 6 to avoid overcounting. Therefore, the total number of ways is (6 x 5) / 2 / 6 = 15.

6. All faces painted with different colours: There are 6 ways to choose the colour for the first face, 5 ways to choose the colour for the second face, and so on. Therefore, the total number of ways is 6 x 5 x 4 x 3 x 2 x 1 = 720.

Adding up the above values, we get a total of 6 + 5 + 25/6 + 10 + 15 + 720 = 781 2/3. Therefore, the correct answer is none of these.