In how many distinct ways can 8 beads of different colours be strung to form a necklace if arrangements that can be obtained from one another by rotation or reflection are considered the same?

Introduction / Context:
This problem is another necklace counting question involving symmetry. We have 8 distinct beads and we are forming a circular necklace, where arrangements that are the same under rotation or reflection are considered identical. This requires using the formula for the number of distinct necklaces with n different beads, taking both rotations and reflections into account.

Given Data / Assumptions:
Number of beads = 8. All beads are different in colour. Beads are arranged in a circular necklace. Rotations of a bead arrangement are considered the same necklace. Reflections (mirror images) of a bead arrangement are also considered the same necklace.

Concept / Approach:
For n distinct beads arranged in a circle, the number of circular permutations when only rotations are identified is (n - 1)!. When reflections are also identified, each unique circular arrangement is counted twice in this count (once for each orientation), so we divide by 2. Therefore, the number of distinct necklaces equals (n - 1)! / 2 when both rotations and reflections are treated as equivalent for n distinct beads.

Step-by-Step Solution:
Step 1: Let n = 8 for the 8 distinct beads. Step 2: First handle rotational symmetry. The number of circular permutations ignoring reflection is (n - 1)! = 7!. Step 3: Compute 7! = 5040. Step 4: Now include reflection symmetry. In a necklace, each arrangement has a mirror image that looks the same when flipped, so every distinct necklace is counted twice in the 7! count. Step 5: Divide by 2 to correct for the double counting: distinct necklaces = 7! / 2. Step 6: Calculate 7! / 2 = 5040 / 2 = 2520. Step 7: Therefore, there are 2520 distinct necklaces.

Verification / Alternative check:
We can test the general formula by using smaller numbers of beads. For instance, with n = 4 distinct beads, the formula gives (4 - 1)! / 2 = 3! / 2 = 3 distinct necklaces, which can be verified by drawing and enumerating possible patterns up to rotation and reflection. This matches known results. Similarly, for n = 3 beads, the formula gives 1 necklace, which is also intuitive. These checks support using (n - 1)! / 2 for n = 8, giving 2520.

Why Other Options Are Wrong:
The value 5040 is 7! and accounts only for rotation but not for reflection symmetry. The value 2880 does not match any standard division of 7! and indicates an incorrect adjustment. The value 4320 is also inconsistent with the required division by 2. The value 1260 is exactly half of 2520 and would be correct only if we had already divided by 2 for another symmetry incorrectly. Only 2520 corresponds to 7! / 2 and properly includes both rotational and reflection equivalence.

Common Pitfalls:
Some students mistakenly use 8! for linear arrangements or 7! for circular arrangements without considering reflections. Others may incorrectly divide by 8 instead of 2, or they apply the reflection correction twice. Another common issue is misunderstanding the phrase arrangements that look the same under rotation and reflection. Always ask whether both types of symmetry are being considered and apply the formula accordingly. For n distinct beads and both symmetries, use (n - 1)! / 2.

Final Answer:
The number of distinct necklaces that can be formed from 8 different beads is 2520.