Combinatorics: Coloring a prism

Question

Given is the following six-sided prism/pyramid structure:

The 13 different sides of the this body should be colored with 20 different colors (including red, blue, green and yellow). In how many ways is this possible if:

(a) There are no constraints?

(b) Each side has to have another color?

(c) The "base" side (Grundfläche) of the structure has to be red?

(d) Two sides are maximally allowed to be red?

(e) The number of red, blue, green and yellow side should be the same?

These are my approaches - feedback on correctness and better ways of solving the tasks is very welcome:

(a) Seems straightforward as we have 20 colors to chose from and 13 sides: ${20}\choose{13}$

(b) Here I would say 20 * ....* 8!

(c) I would say that we have for the other 12 sides 20 colors to chose from, hence: ${20}\choose{12}$, and not really a choice for the 13. side

(d) So, either none of the two remaining sides is red, one of them is red or both are, which would amount to: ${20}\choose{11}$ + 20^2 (none is red) + 2*20 (one is red, 20 choices for the other)

(e) So, either we have each of the four colors once: ${20}\choose{9}$ + ${13}\choose{4}$, or twice: ${20}\choose{5}$ + ${13}\choose{8}$ ... is this approach correct?

Answer 1

Concerning your answers:

(a) $20 \choose 13$ is the number of ways of choosing $13$ different colors from $20$ total, without regard to order chosen. That the colors must be different would be a constraint.

As N.F. Taussig also pointed out, the prism can be turned 6 ways and still be considered the same. That is, if you do two colorings, but can obtain one from the other simply by rotating the prism, they should be considered the same coloring.

(b) The wording here is very, very poor. Perhaps you mean a different color? This then is the constraint you put on your answer to (a). Your solution would be correct if the 13 faces were all distinct, but that is not case here.

(c) You would be correct if you had the correct answer for (a).

(d) Your answer makes no sense, even overlooking the (a) error. There $19$, not $20$, choices for color if red is not included. There three possibilities:

• None of the faces are red. So each face can have any of the $19$ possible colors.
• One face is red. There are $3$ possible choices for this red face. (Can you figure out why it is only $3$?). For the case where the bottom face is red, you have the answer to (c), except with only 19 colors to choose from. For the other 2 cases, you will need to pay attention to the fact that the red face determines an orientation to the prism.
• Two faces are red. If one is the bottom face, then the situation is almost the same as the "one red face not on the bottom" senario, except that now the bottom is no longer a face to be colored. If both are on the sides, you have 3 different cases: both are peak faces, both are mid faces, or one is peak face and the other is a mid face. Then you also have to account for how the two faces align with each other. Once you've counted all the possible ways these can occur, you need to multiply that by number of ways to color the other 11 faces with the other 19 colors.

(e) The number of red,green,blue, yellow sides can also be 0 each, or 3 each. And of course, you need to think harder about how many ways each of these can be done.