sahilchahal wrote:You have a six-sided cube and six cans of paint, each a different color. You may not mix colors of paint. How many distinct ways can you paint the cube using a different color for each side? (If you can reorient a cube to look like another cube, then the two cubes are not distinct.)
(A) 24
(B) 30
(C) 48
(D) 60
(E) 120

Approach 1:
Total number of ways to paint the cube = (total number of ways to arrange the 6 colors)/(total number of ways to orient a cube).

Total number of ways to arrange the 6 colors = 6! = 720.
The total number of ways to orient a cube = 24.
Thus:
Total number of ways to paint the cube = 720/24 = 30.

Approach 2:
Count the number of ways to arrange the other 5 colors RELATIVE to the first color painted.

Once the first color is painted on the BOTTOM FACE, the number of options for the TOP FACE = 5. (Any of the 5 remaining colors.)
The remaining 4 faces form a CIRCLE around the middle of the cube: front face - left face - back face - right face.
The number of ways to arrange n elements in a circle = (n-1)!.
Thus, the number of ways to arrange the remaining 4 colors = (4-1)! = 6.
To combine the options above, we multiply:
5*6 = 30.

The correct answer is B.