Is it possible to find two distinct 4-colorings of the tetrahedron which use exactly one of each color?

Question

I'm working on a problem for Discrete Math, and I'm having some trouble. The question is:

Determine the pattern inventory for red/green/blue/yellow colorings of the vertices of a regular tetrahedron.

(a) How many distinct 4-colorings are there?

(b) Is it possible to find two distinct 4-colorings of the tetrahedron which use exactly one of each color? Explain.

I've already completed part a (and by extension, I have the pattern inventory completed). The answer should be 36.

But I'm having trouble with part b. I'm not sure that I understand the question, or how to work it. Maybe I've overlooked something. Could someone help me out, please?

EDIT: The pattern inventory should be:

1/12[(r+b+g+y)^4 + 8(r+b+g+y)(r^3+b^3+g^3+y^3) + 3((r^2+b^2+g^2+y^2)^2]

Answer 1

For b you need to determine if reflections are different or not. There are four vertices, so if you use all four colors you have one vertex of each color. If you color them on two tetrahedra, put the tetrahedra on a table with the red vertex up, you can have green/blue/yellow go clockwise on one and counterclockwise on the other. These are the same if reflections are allowed and not if they are not.