2000 AMC 12 Problem 25

There are $8!$ ways to assign the eight distinct colors to the eight faces. Two assignments give the same octahedron exactly when one is a rotation of the other. The rotation group of a regular octahedron has $24$ elements. Because all eight colors are different, no nontrivial rotation fixes a coloring, so each distinguishable octahedron corresponds to exactly $24$ assignments. Therefore the number of distinguishable octahedrons is $$\frac{8!}{24} = \frac{40320}{24} = 1680.$$ Thus, the correct answer is E.