Question

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)

$210$

$560$

$840$

$1260$

$1680$

Solution:

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.

2000 AMC 12 Problem 25

Solution:

Problem 25 in Other Years