10. 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 of painting are there? (Painting methods are considered different only if they remain different after any rotation of the cube)
【Key Points】Counting, Combinations
【Difficulty】Moderate
【Answer】 35
【Analysis】(1) Taking 3 colors out of 5 colors, the six faces of the cube are paired, and any 3 colors form one coloring method. There are $C_{5}^{3}=10$ ways;
(2) Taking 4 colors out of 5 colors, for each set of 4 colors, first determine two colors to paint two pairs of opposite faces, and the remaining two colors to paint one pair of opposite faces, there are $C_{5}^{4} C_{4}^{2}=30$ ways;
(3) Taking 5 colors out of 5 colors, first determine 1 color to paint one pair of opposite faces, the remaining 4 colors (denoted as $a$, $b$, $c$, $d$) have $a b c d$, $a c b d$, $a b d c$, 3 coloring methods, there are $C_{5}^{1} \cdot 3=15$ ways; in total, there are $10+30+15=55$ coloring methods.