I was recently given this question on a practice test, and it was intended to be completed in ~2-3 minutes, for someone with intermediate stats/math knowledge:
For a positive integer $n$, you have $n$ red beads, $n$ blue beads, and $n$ orange beads. You would like to make bracelets from these beads. A bracelet consists of any number of beads on a string in a loop, but your string can only fit up to $m = n + 1$ beads. All beads of the same color are identical. How many distinct bracelets can you make?
I can't figure out how to approach this question. In searching, I found this answer referencing Polya's Enumeration Theorem but this definitely is far more advanced knowledge than the practice test assumed. Is there any other way to go about this? It doesn't have to be rigorous, simply a practical method to figure out how many necklaces can be made.
Is there an elegant way to do this for an equal amount of beads?