Ways to seat r out of n people around a table

[changeofplans]

Homework Statement

Find a formula for the number of ways to seat r of n people around a circular table, where seatings are considered the same if every person has the same two neighbors without regard to which side these neighbors are sitting on.

Homework Equations

None

The Attempt at a Solution

I understand the more basic form of this question; in order to seat n people, we have $\frac{n!}{n}$, since there are n! ways to seat n people and we divide out the n repetitions.

From there, we see that the ways to seat r of n people is simply $\frac{P(n, r)}{r}$. So that gives us the first part of the problem.

But I'm not really sure where to go from here. I'm thinking, in this specific problem, we shouldn't be dividing out the r people, since it's explicitly stated that the only seating arrangements that are considered the same are where the same person has the same two neighbors. So I'm thinking, I definitely have the numerator: P(n, r). And.. then I hit a wall.

Any help is appreciated.

 

[vela]

You'd still have to divide by r because if you shift everyone by one seat, they'll still have the same two neighbors. There's still another way to rearrange a permutation so that everyone has the same two neighbors. You need to account for that.

 

[changeofplans]

Okay, that makes sense. So now I have:

$\frac{P(n,r)}{r}$

What about multiplying r by 2, for the mirrored positions?

$\frac{P(n,r)}{2r}$ = $\frac{n!}{\frac{(n-r)!}{2r}}$

 

[vela]

Sounds good.