# Total number of ways to paint the faces of a regular icosahedron with $20$ distinct colors

If all the 20 faces of a regular icosahedron are painted with a set of 20 distinct colours then the total number of such icosahera possible. The cube analogue of this is more well known and the answer to it is 30.

• Does "distinct colors" mean that we are not allowed to use the same color on two different faces? – lisyarus Jan 18 at 15:13

Just like the cube version - take the symmetric group on the colors (order $$20!$$) and then divide by the rotations of the regular icosahedron ($$A_5$$, order $$60$$). Overall, that's $$\frac{20!}{60}=\frac{19!}{3}$$.