Each face of a cube can be painted in black or white colours. In how many different ways can the cube be painted?

The correct option is 10.

To find the number of different ways to paint the faces of a cube in black or white colors (where two colorings are considered the same if one can be rotated to match the other), we can analyze the cases based on the number of black faces.

Let the total number of faces on a cube be 6. Each face can be painted either black (B) or white (W). Since the problem is symmetric with respect to swapping black and white, we can group the cases by the number of black faces, from 0 to 6:

1. 0 Black faces (and 6 White faces):
There is only 1 way to paint the cube all white.
Number of ways = 1

2. 1 Black face (and 5 White faces):
Since all faces of a cube are identical under rotation, placing a single black face anywhere on the cube results in the same configuration.
Number of ways = 1

3. 2 Black faces (and 4 White faces):
Two black faces can either be:
- Adjacent to each other (sharing an edge).
- Opposite to each other (on opposite sides of the cube).
These two configurations are distinct and cannot be rotated into one another.
Number of ways = 2

4. 3 Black faces (and 3 White faces):
Three black faces can either:
- Meet at a single corner (vertex) of the cube, sharing a common vertex.
- Not meet at a single corner (meaning they lie in a "U-shape" or a line around the cube, where two are opposite and the third is adjacent to both).
These two configurations are distinct.
Number of ways = 2

5. 4 Black faces (and 2 White faces):
By symmetry, this is equivalent to the case of having 2 black faces (just swap the roles of black and white).
Number of ways = 2

6. 5 Black faces (and 1 White face):
By symmetry, this is equivalent to the case of having 1 black face.
Number of ways = 1

7. 6 Black faces (and 0 White faces):
By symmetry, this is equivalent to the case of having 0 black faces (all black).
Number of ways = 1

Summing up all the ways from each case:
Total ways = 1 (0 B) + 1 (1 B) + 2 (2 B) + 2 (3 B) + 2 (4 B) + 1 (5 B) + 1 (6 B)
Total ways = 1 + 1 + 2 + 2 + 2 + 1 + 1 = 10

Thus, there are 10 different ways to paint the cube.