# Coloring the faces of a regular tetrahedron with 3 colors using Burnside's lemma

I have to determine the number of ways the faces of a regular tetrahedron can be colored with three colors. I already know it is 15, but apparently the math professor did not find the solution to his liking. I need to use group theory and Burnside's lemma to explain how you determine the number of ways the faces of regular tetrahedron can colored with three colors. What I understand: the basics of abstract algebra/group theory, but not much about how to apply group theory to solve this specific problem.

• Do you have to use all three of the available colors, or do you include colorings where only one or two of the colors is used? – bof Oct 9 '16 at 6:01
• @bof It seems that only rotations are allowed and not all the colours have to be used. – Parcly Taxel Oct 9 '16 at 7:13
• This MSE link here uses PET but you should be able to derive Burnside from it since the terms in the cycle index also let you calculate the number of colorings fixed by a permutation having that factorization into disjoint cycles. Use inclusion-exclusion for exactly three colors as opposed to at most three colors. – Marko Riedel Oct 9 '16 at 20:28