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

Question

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.

Answer 1

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 $\frac{20!}{60}=\frac{19!}{3}$.