Exercise 11. A palette of 6 different colors is given. In how many ways can a cube be painted, using all 6 colors, and exactly one color per face? Two ways of coloring are considered identical if one can be obtained from the other by any rotation in space.
Solution to Exercise 11 We place the face with color 1 at the bottom. There are 5 choices for the top face. There are then 24 choices for the four side faces, but considering rotations, there are only 6 possible choices, so the answer is $5 \times 6=30$.