10. (15 points) Use five different colors to paint the six faces of a cube. If two adjacent faces cannot be painted the same color, how many different ways are there to paint the cube? (Painting methods that are the same after any rotation of the cube are considered the same)


【Analysis】Use five different colors to paint the six faces of a cube. First, determine 1 color to paint a pair of opposite faces, and the remaining 4 colors (denoted as $a, b, c, d$) have $a b c d, a c d b, a c b d$, 3 coloring methods, which results in $C_{5}^{1} \cdot 3=15$ ways; solve accordingly.

【Solution】Solution: According to the analysis,
$$
C_{5}^{1} \cdot 3=5 \times 3=15 \text { (ways); }
$$

Answer: There are 15 different coloring methods.