Find the number of arrangements

Question

There are 9 people and they have to sit around a round table, find the number of arrangements so that in each arrangement no person has the same neighbour ?

Answer 1

I'm guessing this is asking for a Hamilton decompostion of $K_9$, i.e., a decomposition of $K_9$ into $C_9$'s. This can be achieved using the Walecki decomposition. As in my answer here Partition edges of complete graph into paths of distinct length it can be achieved by:

Here's another example, drawn to show how people sit around the table: