Example 1 Choose several colors from 6 different given colors to color a cube, with each face colored with one color and ensuring that any two adjacent faces sharing a common edge are of different colors. How many different coloring schemes are there (if one of the two colored cubes can be rotated to match the coloring state of the other, then the original coloring schemes of the two are considered the same)?
(1996, National High School Mathematics Competition)
Explanation: Clearly, the maximum number of colors used is 6, and the minimum is 3. Below, we use the grouping method to count.

(1) When using 6 colors, each color is used to paint one face. According to the problem, the first color can be placed on top by flipping. At this point, the bottom is one of the remaining 5 colors, giving 5 different coloring methods. The other 4 colors are used to paint the front, back, left, and right faces. We can choose one of these 4 colors to paint the front face. Then, the back face is one of the remaining 3 colors, giving 3 different coloring methods. The remaining 2 colors are used to paint the left and right faces, giving 2 different coloring methods. Therefore, the total number of different coloring methods is $5 \times 3 \times 2 = 30$.

(2) When using 5 colors, we select 5 colors from the 6 available colors, and then choose one of these 5 colors to paint two opposite faces, which can be considered as the top and bottom faces, giving $6 \times 5 = 30$ different coloring methods. The remaining 4 colors are used to paint the 4 side faces. We choose one of these 4 colors to paint the front face. Then, the back face is one of the remaining 3 colors, giving 3 different coloring methods. The remaining 2 colors are used to paint the left and right faces. Unlike in (1), at this point, the cube can be rotated 180 degrees around the axis through the centers of the front and back faces, so there is only 1 coloring method. Therefore, the total number of different coloring methods is 90.

(3) When using 4 colors, we select 4 colors from the 6 available colors, and then choose 2 of these 4 colors to paint two opposite faces. The number of different coloring methods is $\mathrm{C}_{6}^{4} \mathrm{C}_{4}^{2} = 90$.

(4) When using 3 colors, the number of different coloring methods is $\mathrm{C}_{6}^{3} = 20$.

In summary, the total number of different coloring methods is 230.

Comment: Although this problem is just a fill-in-the-blank question, it is not easy to solve correctly. The standard answer given at the time was 320, which is incorrect. The reason for the error lies in (2), where the last 2 colors are used to paint the two faces. Since the cube can be rotated, there is only 1 method, but the standard answer incorrectly considered it as 2 methods.