Question

Six colors of paint are available. Each face of a cube is to be painted a different color. In how many different ways can this be done if two colorings are considered the same when one can be obtained from the other by rotating the cube?

• A. $5$
• B. $6$
• C. $30$
• D. $6!$

Answer 1

The correct option is C $30$

A cube can be rotated into $6\times 4=24$ configurations (i.e. the red face can be any one of the 6, and then there are 4 ways to rotate it that keeps that face red)
So the number of different colourings (counting rotations, but not mirror reflections, as the same) is $\frac{6!}{24}=30$