circular r-permutations of n

Question

My book, Discrete Mathematics And Its Applications by K.H.Rosen asks to find a formula for circular r-permutations of n people. That is, sitting of r of these people around a table when two sittings are considered equal if they look the same by rotation of the table.

As I searched, the answer must be, P(n,r)/2r but the book says something different:

The books answer.

So my problem is why? Why do we divide by r and not by 2 at the end? And why didn't "design the head" step divide the answer by r?

Answer 1

What you probably got the answer is the "bracelet problem", what is being asked is the "necklace problem". The difference is the bracelet is assumed to be equivalent under reflection, whereas necklace is not. see wikipedia article here

Answer 2

I know this question was asked about 3 years ago, but it is never too late if the outcome is good!

Following what Muralidharan said in the comment section, start with the simplest possible case: in how many ways we can rearrange $r$ people in a circular table with $r$ chairs? Two sittings are considered equal if they look the same by rotation of the table, so we can apply this consideration sitting one of them and permute the rest. Therefore, we will have $$ P_\text{circ}(r, r) = (r-1)! = \frac{r!}{r} $$ different forms to rearrange them, where $P_\text{circ}(n, r)$ stands for a circular $r$-permutation over $n$ elements.

If we have to sit $r$ out of $n$ people, we will have $C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$ ways to choose $r$ of them and $P_\text{circ}(r, r) = (r-1)!$ ways to sit them. Thus, following the product rule, $$ P_\text{circ}(n, r) = C(n, r)P_\text{circ}(r, r) = \frac{n!}{r!(n-r)!}(r-1)! = \frac{n!}{r(n-r)!} = \frac{P(n, r)}{r}, \tag{1}\label{eq:circ} $$ where $P(n, r)$ stands for a $r$-permutation over $n$ elements.

If we want to be more strict and treat as equal rearrangements those where all people have the same neighbours no matter the orientation (reflection symmetry), for the $n\ge3$ cases you will need to divide by 2 \eqref{eq:circ}. Notice that it will not be relevant to sit one person and rearrange the rest of them in a clockwise or counterclockwise orientation.