Question about coloring sheets of paper

Question

Consider 3 sheets of paper. Each of them can be colored in 1 of 6 colors. If you pay attention to symmetry, how many ways are there to color them? By that I mean (red, red, green) would be the same as (green, red, red) but wouldnt be the same as (red, green, red). I think the solution is 126 but im not sure about that.

Answer 1

Split the possibilities up into:

3 different colours These can be chosen in $6C3=20$ ways. They can then be arranged in 3 ways according to your symmetry rule.

2 different colours The one used twice can be chosen in $6$ ways and the other colour can be chosen in a further $5$ ways, giving a total of $30$. They can then be arranged in 2 ways.

1 colour This can be chosen in $6$ ways.

The total is indeed $126$.

Answer 2

• If the first one is 1, then the last one can be 1 solely.
• If the first one is 2, then the last one can be 1 or 2.
• If the first one is 3, then the last one can be 1 or 2 or 3.
• etc.

Thus the choices for the pairing of the first and last one amounts into 1+2+3+4+5+6=21 cases.

Each one can be adjoint with any of the six colors for the middle one.

Thus in total 21*6=126 cases.

--- rk