How to find different colorings of cubes with restrictions When painting a cube, say 1 side is painted color a, 2 are painted color b, and 3 are painted color c. How can I find the number of possible ways it can be painted? Rotations ARE the same configuration.

I'm going to use red, green, and blue as my colors (since that feels more natural to me). One side needs to be red, two sides are going to be green, and three sides are going to be blue (since I like blue, I want the cube to be mostly blue).

I'm going to start by paining one face red. Next, I am going to paint the opposite face. I have two choices: I can paint that face either green, or blue.

1. If I paint the opposite face green, then I have to paint one of the remaining face green, and the other three remaining face blue. There is only one way to do this.

2. If I paint the opposite face blue, then I have to color two of the remaining faces green. There are two ways to do this: either I paint two adjacent faces green, or I paint two opposite faces green.




As these two cases exhaust all possibilities, there are a total of three different ways of painting a cube subject to the constraints you gave.