Necklace/chain arrangements with 5 distinct beads: How many distinct chains (flip considered identical) can be formed from 5 different colored beads?

Introduction / Context:
Counting circular bead arrangements where reflections (flips) are considered identical uses necklace or “free” circular permutations. For n distinct beads on a loop with flips identical, the count is (n − 1)! / 2 for n ≥ 3 (when no additional symmetries collide distinct colorings in general position).

Given Data / Assumptions:

• n = 5 distinct colored beads.
• Rotations considered the same; reflections (mirror images) also considered the same.

Concept / Approach:

• Free necklace count for distinct beads: (n − 1)! / 2.

Step-by-Step Solution:

Verification / Alternative check:
Polya’s enumeration for all-distinct colors also yields (n−1)!/2 when flips are identified, matching 12 for n=5.

Why Other Options Are Wrong:

• 24 equals circular without identifying flips; 18 and 30 are not standard counts here.

Common Pitfalls:

• Forgetting to divide by 2 to identify mirror images as the same.

Final Answer:
12