Expert: Scott A Wilson - 7/6/2011

Question
This question is really confusing for me:

In how many ways can the 6 surfaces of a cube be coloured using 6 colours?

Answer
To do this, lets number the colors 1, 2, 3, 4, 5, and 6.
Note that no matter which way the cube is colored, oolor 1 can be put on top.

Given this, there are 5 other colors that could be placed on the bottom,
so we're up to 5 choices.

Given one on the top and one on the bottom, out of the 4 remain colors,
the next side colored will be turned toward the front.
This could be done to any side.  The color chosen will be on one of the sides eventually,
so pick that as the next color to look at and put it in front.
This again adds no choices, so with 3 sides colored,
there are 5 ways to do it and there are 3 colors left.

For the side to the left of the front, there are 3 choices.
Once this has been done, there are 2 choices for the back.
The final step is to apply the only color left to the side on the right.

Reviewing what I've written, there was 5 choices for the bottom
and 3*2*1 choices for the sides, which is 5*3*2*1 = 30.