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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 ?

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closed as off-topic by Rohan, JonMark Perry, Namaste, zhoraster, Antonios-Alexandros Robotis Feb 20 '17 at 16:53

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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:

Decomposition of K_9 into C_9

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

Decomposition of K_9 into C_9

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