How many distinguishable ways are there to paint the six faces of a cube, each a different one of six given colors? (Two colorings are the same if one can be rotated to match the other.)
Fix a labeling of the $6$ faces of the cube; there are $6!$ ways to assign the $6$ distinct colors to them.
The rotation group of the cube also has order $24$.
With six different colors no rotation except the identity fixes a coloring, so each distinguishable cube corresponds to $24$ colorings, and the number of distinguishable colorings is $6!/24=30$.
