How many distinct colorings can be obtained when using 6 different colors to color the faces of a 3d-cube (one face - one color)? two colorings are identical if they can be obtained by rotation (but not reflection).

30 distinct colorings

Step-by-step explanation:

First, we need to calculate the total number of orientations that any unique cube can have. To do this, we fix the top face. As any face can occupy the place of the top face, there are 6 possible options for top face. Now, we fix the front face. Any face of the 4 faces of the side of the cube can occupy that place. This would give us a total 0f 6*4 = 24 possibles orientations that any particular cube can have.

Now, we need to calculate the amount of ways to paint the cube that we have. We paint one of the faces, to do this we have 6 possible colors to choose. Now, to paint any other face, we have 5 colors, then 4 colors, and so on. The total number of ways will be 6*5*4*3*2*1 = 6! = 720.

Finnally, as we have established that any unique cube will have 24 possible orientations, the amount of possible unique cubes multiplied by the 24 possibles orientations should give us the total number of ways to paint the cube:

24* n = 720

Said in another way, if we divide 720 by 24, we should get the possible amount of unique cubes we can have. This would be:

n = 720/24 = 30