18.2 Arranging people in a circle

Sometimes we want to arrange people in a circle. If you ever feel the need to procrastinate over the organising of a dinner party for $n$ people, you can consider the number of unique ways in which you can seat your guests.

The key thing to note, is that a circle is "just" a straight line, where we have joined the ends. This means that the people at either end of the line now sit next to each other. We can draw a handy diagram²² 2 This usually helps with almost any maths problem:

Note that there is no way to have a "start" of a circle (whereas we can clearly have a front and back of a line). Because rotating people doesn’t actually change their arrangements relative to each other, we can say that (for example, all of these arrangements would be the same)

• •
  ​

  123456

• •
  ​

  234561

• •
  ​

  345612

• •
  ​

  456123

• •
  ​

  561234

• •
  ​

  612345

If we draw out some of these arrangements, it’s clear that the position of the people (again relative to each other, not the "start" because a circle doesn’t have a start) hasn’t changed. Person 1 is still next to Person 2 and 6, Person 2 is still next to Person 1 and 3 (and so on).

How many ways can we arrange people around a table, though? Well, when we were looking at the number of ways we could arrange people in a straight line, we were thinking about this relative to the start and end of the line. We don’t have any handy features of geography ³³ 3 Fun fact: the Austrian foreign minister once described Italy as a ”mere feature of geography” to define the arrangement of the people at the table in relation to.

The way we can get out of this pickle is by defining one! Let’s pick a person, and then think about how we can seat everyone else at the table relative to that person. For $n$ people, we can seat the first person at an arbitrary seat (which there is one way of doing). We can then work our way round the circle. If we pick a direction (left or right) to mean "next", we can then establish that there are $n-1$ options for the person immediately next to the first person. There are then $n-2$ options for the person two spaces away, $n-3$ for the person three spaces away, and so on. By the time we’re back to the person on the other side of the first person, we have only one choice. Overall, this means that: