Number of Ways of Seating People at Circular Table

Theorem

Let there be $n$ people to be seated at a round table.

Let $N$ be the number of different ways to seat those $n$ people.

Note that if person $A$ is seated between person $B$ and person $C$, then having $B$ on the left and $C$ on the right is considered different from having $B$ on the right and $C$ on the left.

Then:

$N = \paren {n - 1}!$

Proof

From Number of Permutations of All Elements, the number of different ways of arranging $n$ people in line is $n!$.

However, a round table has no beginning and end.

Hence the vital factor is the arrangement relative to an arbitrary given person.

So we fix one person, and arrange the other $n - 1$ people relative to that person.

From Number of Permutations of All Elements, there are $\paren {n - 1}!$ ways to arrange those people.

$\blacksquare$

Examples

$8$ People

$8$ people can be seated at a circular table in $5040$ ways.

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: The number of ways of seating $n$ people at a circular table