Please tell me the total number of permutations possible of the beads in a necklace where all the beads are distinct. The necklace consists of n distinct beads.
Answer as per me: the answer is $(n-1)!\:$, as the clockwise and anti-clockwise arrangements are different.
Answer given in various online courses: The clockwise arrangement of the necklace shown is B1-B2-B3-B4. Since the necklace can be flipped, the total permutations = (n-1)!/2. Reason: After flipping the necklace, the anti-clockwise arrangement is the same again i.e. B1-B2-B3-B4. The link to these online courses are:
1) https://www.askiitians.com/iit-jee-algebra/permutations-and-combinations/circular-permutations.aspx
2) https://www.youtube.com/watch?v=6BoAwmUlfqs>
The 2nd link is a video lecture in the language of Hindi. Start this video from 13:05 minutes, so as to avoid the unnecessary things.
Can you tell me which one is correct with an appropriate explanation.
It's only when one allows the necklace to be flipped over that division by 2 is needed. If one keeps the necklace say on a table and only rotates it (no fair picking necklace up in space) it's the $(n-1)!$ answer.