# 62 Colors

Discrete Mathematics Level pending

This is the Great Rhombicosidodecahedron (one of the Archimedean Solids):

In this picture it is colored using $$3$$ different colors.

However, suppose you have $$62$$ different colors, and you wish to paint each of its $$62$$ faces in a different color.

Let $$\dfrac{a}{b}$$ be the ratio of the number of unique ways to paint this solid divided by the total number of ways this solid can be painted, where $$a$$ and $$b$$ are coprime positive integers.

What is $$a+b$$?

Note: Unique implies that for two coloring combinations $$A$$ and $$B$$, you can't pick up $$A$$ and reorient it to make it look exactly like $$B$$.

###### Image credit: commons.wikimedia.org
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