So, I am "relearning" about Combinatorics. I just started with "An introduction to combinatorics and graph theory (David Guichard)", and I'm really confused regarding one of the examples
EXAMPLE 1.2.4: Imagine that we have three dice that can change color: one can be red or white, one green or white, one blue or white. If they have colors red, green, and blue, there are P(6; 3) = 6*5*4 possible outcomes. If we arrange the dice in all 120 possible ways and take a picture, we get 120 obviously different photographs. (...)
In this particular example, what "possible outcomes" refers to? I expected all possible outcomes to be 6^3 = 216, as the arrangement seems to matter due to the different colours, and no explicit requirement regarding repetition in values is presented. In my opinion, P(6;3) represents the outcomes of a roll where the three dice have different "numbers".
Am I misunderstanding something?
If the dice are allowed to have the same number, the correct number of permutations is $6^3=216$, just as you surmised.
On the other hand, were the dice not allowed this property, we'd get $6C3$ or $\frac{6!}{3!3!}$ which is the $20$ eventually stated. The $120$ is $6P3$, the simple difference between the two is that $6P3$ will allow $1,3,6$ and $6,1,3$ to both be counted, as well as any other reshuffles of those three digits, whereas $6C3$ will only include $1,3,6$.