Question

In how many ways can $9$ persons sit around a table so that all shall not have the same neighbours in any two arrangements?

Answer 1

Persons can sit at a round table in

$(9-1)!=8!=40320$ways.

But in clockwise and anti-clockwise arrangement, every person will have the same neighbours.

So, the required numbers of ways are

$\frac{1}{2}\times 8!=\frac{40320}{2}=20160$ways.