Quick way to count circular arrangements when two people cannot sit together

Suppose (n) people are seated around a circular table, and two specific people—say A and B—must not be adjacent.

First, count all possible circular arrangements:

(n-1)!

Now count the arrangements where A and B sit together.

Treat A and B as one block. We now have (n-1) objects around the table, giving:

(n-2)!

A and B can switch places inside their block, so multiply by 2:
2(n-2)!

Subtract the unwanted arrangements from the total:
(n-1)!-2(n-2)!

Factorising gives: {(n-3)(n-2)!}

This works for (n>3), assuming rotations are considered identical but clockwise and anticlockwise arrangements are different.

The main idea is simple:

Valid arrangements = All arrangements − Arrangements where the two people sit together.