Consider a bracelet of 5 colored beads. How many different bracelets are there if all the 5 colors are different? Take into account rotations and flipping. Verify the correctness of your solution considering the same problem for a bracelet of 4 beads with 4 different colors.
My answer is : There are possible (n-1)!/2 bracelets for n distinct colors, like in our case. Thus for n=5, there are possible 4!/2=12 different bracelets. I think that I'm missing something.
I would like to know to deal with such problems.
As @Doc notes in a comment, Polya enumeration is a way to solve this kind of question. But the particular questions you ask can be answered without it and in fact your answer looks good to me. We can always rotate the bracelet so the red bead is at the top. That leaves $(n-1)!$ ways to place the other beads, but they come in pairs where you get the one from the other by flipping in the vertical axis. So, $${(n-1)!\over2}$$
Why do you think you are missing something?