I have a set of N identical Platonic (regular) solids, each painted with a different color on each face--the same set of colors for each of them. If I choose at random one face of each of the solids, the probability that all the chosen faces are of different colors is 2/3 the probability that there are three of one color and the rest all different colors.

How many of these solids do I have, and how many faces are on each?