Question

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.

Short answer

\(n!/((n - r!)2r)\) if \(r > 2\) and \(n!/((n - r)!r)\) if \(r \le 3\)

Step-by-step solution

Definition of permutation

Definition permutation (order is important): \(P(n,r) = \frac{{n!}}{{(n - r)!}}\)

Use permutation to find the formula

The equivalent of numbering all the seats around the table and considering how many ways we can assign\(r\)people to specific seat numbers is:\({\rm{n}}!/({\rm{n}} - {\rm{r}})!\).

Divide by a couple of factors: One is\(r\), because there are\(r\)positions at the table from which we can start listing any permutation, so permutations here which are equivalent.

The other divisor is\(2\), because there's another set of permutations which consist of the same seating in the opposite direction (clockwise vs. counterclockwise).

But this divisor only applies if\(r > 2\), because it takes at least\(3\)people seated around a table to create a situation in which direction makes a difference in the order.

So our formula is \(n!/((n - r!)2r)\)if \(r > 2\) and \(n!/((n - r)!r)\) if\(r \le 3\).