Suppose we have the colors 1 thru 6.
Each color is on exactly one face.

To determine the number of possibilities we rotate the cube so that color 1 is on the bottom face.
The top face can be any of the remaining 5 colors.
We know rotate the cube so that color 2 is on the left face (and color 1 still at the bottom).
If color 2 is at the top face we rotate so that color 3 is the left face.
Rotating this way doesn’t change the number of possibilities.

For the front, right and back face we have 3 colors left. And there are 6 ways they can be arranged (3x2x1).

The total number of unique ways to paint the cube is 30 (5×6).

There are six distinct colors and the cube has also 6 faces. that means it is not possible that two faces have the same color!! So if we number the colors 1-6 and the order of the faces will be: top, bottom, left, front, right, back and put the colors in this order than we have 6! =720 ways. But after we choose color 1 for the top there are 5 colors for the bottom. we are left with 4 colors that cab be arranged in 4! ways but each of them can be rotated 4 times to we actually have only 4!/4=6 unique ways. there are different ways for the top and bottom colors but the cube can be flipped on 3  different axis so we have 15*6/3=30 ways in total.