# The number of arrangements when twelve people sit in circle such that two certain people cannot be together

There are twelve people, arranged in a circle. Two people, for example, Mike and Ike, can not be together.

This is what I have done:

$$\frac{12!}{12}-\frac{11!2!}{12}$$

The first section representing the total arrangements in a circle, and the second thinking Mike and Ike as an independent set ($2!$), resulting in $11!2!$, representing the arrangements between them and the original group. That gives me: $33,264,000$, but the answer (unsure if correct) gives me: $32,659,200$. Assuming the provided answer is correct, what have I done wrong?

Your logic is good however you made a mistake in calculating the number of arrangements where mike and ike are seated next to one another.

Instead of "dividing by symmetry" (which is how many people are taught how to approach these circular arrangement problems, and is quite useful for situations which have a great deal of symmetry), let us instead count the number of circular arrangements where mike and ike are seated next to one another directly via multiplication principle.

• Step one: Let Mike sit in any chair at the table. Such a choice does not matter since we consider all rotations of the circle to be identical. (convince yourself of this)
• Step two: Ike will sit next to Mike. Pick which seat Ike sits in. He has two options, namely to the right or to the left of Mike. $[2~\text{options}]$
• Steps three and on: For each remaining open seat, select someone to sit in it. $[10!~\text{options}]$

This gives a total count of $2\cdot 10!$ number of "bad scenarios" where Mike and Ike sit next to one another, giving the correct total of arrangements which avoid this being $11!-2\cdot 10! = 32659200$

What went wrong with your calculations?

When "dividing by the symmetry of the circle," you neglected to take into account that the circle "has eleven points" around it when counting the scenarios where Mike and Ike are next to one another, since you have Mike and Ike occupying a single point. As such, you should have divided by eleven, not by twelve.