Circular Permutation, clockwise and counterclockwise can't be distinguished?

Question

I did a couple of problems on circular permutation, where clockwise and counterclockwise were distinguishable.

My textbook says if they aren't distinguishable, then it's $(n-1)!/2$ ... Well this makes sense. But then the problem went like, if 10 "different" beads are to be used to make a necklace, then the number of necklaces that can me made are (10-1)!/2

Wait now hold on! If all of those beads are different, I think this should be enough of a reason to think the CW and AW can be distinguished. So shouldn't it be simply (10-1)!= 9! ..?

I could understand if beads were all alike.

If my textbook's right, then could you explain me why? Also, what would you do, if the condition was all beads are alike?

Answer 1

Your textbook is right because you can turn the necklace over giving the same necklace, but with the beads essentially reversed. If all the beads are indistinguishable, well there's easily only 1 permutation.

Answer 2

Clockwise and counterclockwise can be "indistinguishable" in the same way that rotations can be "indistinguishable".

If a necklace with ten different-colored beads were placed in a circular shape in front of you, I think you could see which colors of beads were closest to you. If you closed your eyes for a moment, and then someone rotated the necklace around its circle so that those beads were now furthest from you, I think you could tell that something had changed when you opened your eyes. Am I wrong about either of those statements?

When the authors of the book wrote the word "indistinguishable," they were not saying anything about your ability to detect when the necklace is rotated or reversed in direction. If the necklace has ten different beads, then of course you can tell when the necklace is rotated or flipped! What the authors meant by "indistinguishable" is merely that it is still the same necklace, and does not become a new arrangement of beads on a necklace every time you move it.

Another way to look at it is that if I make a necklace by stringing all the beads in a clockwise order, and then I make another necklace by taking the beads in the same sequence of colors but this time I string them in counterclockwise order, the two necklaces I have just made are considered identical. We do not care which way I went around the circle when I strung the beads, because we can always flip the necklace over and now it looks just like it would if I had strung it in the other direction. That is the sense in which the clockwise and counterclockwise sequences are "indistinguishable."