A cube is painted so that one side is blue, two sides are red, and three sides are green. How many different such cubes can be painted? Two cubes are considered the same if one cube can be rotated in any way to match the second cube.
We can count.
Fix the blue side of the cube to be on top.
The 2 reds can be
1 on bottom and 1 on the side.
1 on the side and the next adjacent to it.
1 on the side and 1 opposite it.
The greens fill up all the other sides.
All other possibilities can be rotated to one of these.
So there are 3 distinct cubes.