Determine the number of ways the faces of a regular tetrahed

Determine the number of ways the faces of a regular tetrahedron can be colored with three colors. Show all of your work and adequately explain your reasoning for your solution.

Solution

A tetrahedron has 4 sides. The ratio of the number of faces with each color must be one of the following:

4 : 0 : 0 , 3 : 1 : 0 , 2 : 2 : 0 , or 2 : 1 : 1

The first ratio yields 3 appearances, one of each color.

The second ratio yields 3 . 2 = 6 appearances, three choices for the first color, and two choices for the second.

The third ratio yields 3 appearances since the two colors are interchangeable.

The fourth ratio yields 3 appearances. There are three choices for the first color, and since the second two colors are interchangeable, there is only one distinguishable pair that fits them.

The total is 3 + 6 + 3 + 3 = 15 appearances