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In the case of a ring or necklace, we divide all possible outcomes by 2. Why not in the Round table case for instance, if we have $5$ distinct key rings on a table, all we have to do is $\frac{(5-1)!}{2}$ then why we can't divide by $2$ in the case of a round table like if we have $5$ people to arranged. Why we can't divide $(5-1)!$ by $2$?

Napoleon Bonaparte
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Sobia khalid
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Let's start with your example of a necklace.

Imagine holding a necklace in front of you such that you can see it from one side and someone else, (say, your friend) can see it from the other side.

It's the same necklace with the same arrangement of beads. But the two of you observe different arrangements because of your position relative to the necklace.

And you don't even need another observer. Put the necklace down spread out on a flat surface. What you see is one arrangement. Now, flip it over and you will see the opposite arrangement (mirror image).

So, each permutation contains in itself $2$ permutations because of the fact that there are two ways to view the necklace and they present different permutations.

You can't practically do this with a group of people sitting on a round table. You can't flip them around (the table, chairs, and people, all upside down).$^{[1]}$

That's why we divide by $2$ in case of a necklace (and other similar cases) but not with most cases of circular arrangements (like the table example).

As for your "$5$ distinct key rings on a table" example, I wouldn't divide by $2$ there. That's more like the people on a round table example.

$^{[1]}$ There's a little bit more going on here. Even if you could flip the whole group of people, it still doesn't really give you another arrangement. Let me explain.

People are sentient beings. It's not just about the external observer. There's also a perspective of those sitting.

Say, in one arrangement, B is sitting on the right of A and C on the left. Even if we turn the group upside down, this order (the perception of the people within the group) won't change.

Haris
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