Example 7 Select several colors from the given six different colors to color the six faces of a cube, with each face being colored with exactly one color, and any two faces sharing a common edge must be colored with different colors. How many different coloring schemes are there?
(Note: If we color two identical cubes and can make the colors of the corresponding top, bottom, left, right, front, and back faces of the two cubes the same through appropriate rotation, then we say that the two cubes have the same coloring scheme.)
Solving: Since two faces sharing a common vertex must be of different colors, at least 3 colors are required. Below, we divide the problem into 4 cases.
(1) When all 6 colors are used, first fix one face with a certain color facing up. There are \( C_5^1 \) ways to choose 1 color from the remaining 5 colors to color the bottom face. The remaining 4 colors are used to color the remaining four side faces (a circular permutation of 4 colors), which has \((4-1)!\) ways. Therefore, when all 6 colors are used, the number of coloring schemes is \( C_5^1 \cdot (4-1)! = 30 \).
(2) When exactly 5 colors are used. There are \( C_6^5 \) ways to choose 5 colors from 6 colors. At this point, there must be one pair of opposite faces with the same color. Choose 1 color from the 5 colors to color this pair of opposite faces, and place them facing up and down, which has \( C_5^1 \) ways. The remaining 4 colors are used to color the remaining four side faces (a necklace permutation of 4 different colored beads), which has \(\frac{(4-1)!}{2}\) ways. Therefore, when exactly 5 colors are used, the number of coloring schemes is
\[
C_6^5 \cdot C_5^1 \cdot \frac{1}{2}(4-1)! = 90 \text{ ways.}
\]
(3) When exactly 4 colors are used, there are \( C_6^4 \) ways to choose 4 colors from 6 colors. At this point, there must be 2 pairs of opposite faces with the same color, and one pair of opposite faces with different colors. Place the pair of opposite faces with different colors facing up and down, and choose 2 colors from the 4 colors to color the top and bottom faces (at this point, the top and bottom faces are indistinguishable), which has \( C_4^2 \) ways. The remaining 2 colors are used to color the remaining four side faces so that the 2 pairs of opposite faces have the same color (this can be seen as a necklace permutation of 2 colored beads, or a circular permutation), which has only 1 way. Therefore, when exactly 4 colors are used, the number of coloring schemes is \( C_6^4 \cdot C_4^2 \cdot 1 = 90 \) ways.
(4) When exactly 3 colors are used, there are \( C_6^3 \) ways to choose 3 colors from 6 colors. At this point, all 3 pairs of opposite faces have the same color, and using 3 colors to color them has only 1 way. Therefore, when exactly 3 colors are used, the number of coloring schemes is \( C_6^3 = 20 \) ways.
In summary, the total number of different coloring schemes is \( 30 + 90 + 90 + 20 = 230 \) ways.