Correct Option (2)

A cube possesses 6 faces, and each face can be painted in one of two colours: black or white. Without considering rotational symmetry, the total number of ways to paint the cube would be 2⁶ = 64. However, the question requires counting distinct ways, implying that arrangements that can be rotated into each other are considered identical.

To determine the number of distinct colourings under rotation, we categorize the possibilities based on the number of faces painted black (or white, due to symmetry):

0 black faces (all white): 1 distinct way.
1 black face: 1 distinct way (all positions are rotationally equivalent).
2 black faces: 2 distinct ways (either the two black faces are adjacent, or they are opposite).
3 black faces: 2 distinct ways (either the three black faces meet at a corner, or they form a 'band' or 'line' across the cube).
4 black faces: 2 distinct ways (this is symmetric to 2 white faces, which can be adjacent or opposite).
5 black faces: 1 distinct way (this is symmetric to 1 white face).
6 black faces (all black): 1 distinct way.

Summing these distinct possibilities yields the total number of ways:

1 (for 0 black) + 1 (for 1 black) + 2 (for 2 black) + 2 (for 3 black) + 2 (for 4 black) + 1 (for 5 black) + 1 (for 6 black) = 10 distinct ways.

Incorrect Options:

Options 1 (9), 3 (11), and 4 (12) are incorrect because they do not accurately represent the total number of distinct ways to paint a cube's faces when accounting for rotational symmetry. These values either undercount or miscalculate the unique arrangements possible under the given conditions, failing to apply the principles of combinatorial enumeration with symmetry correctly.