How many different beads can be manufactured with 3 colors?

Question

The question is the following: "A bracelet is made from five beads mounted on a circular wire. How many different brecelets can we manufacture if we have red blue and yellow beads at our disposal."

I know that I have to use the counting theorem, but i can't figure out the details, I do think that the group is isomorphic to $D_5$ but I am not entirely sure about that

Edit: If the group is isomorphic to $D_5$ there are the conjugation classes $\{e\}, \{r,r^4\}, \{r^2,r^3\}, \{s,rs,r^2s,r^3s,r^4s\}$, which each (respectively) leave this amount fixed $3^5,3,3,3^3$ which acording to the orbit counting theorem should be $\frac{1}{10}(3^5 + 3 \cdot2+ 3\cdot2+3^3\cdot 5) = 39$. Could somebody correct me if I'm wrong (altough this is the same awnser as Henry's).

Answer 1

I would guess with rotation and turning over, the patterns could be:

• xxxxx $3$ ways
• xxxxy $6$ ways
• xxxyy $6$ ways
• xxyxy $6$ ways
• xxxyz $3$ ways
• xxyxz $3$ ways
• xxyyz $3$ ways
• xyxyz $3$ ways
• xyyxz $6$ ways

making $39$, but I may have missed some