Question

Find the number of ways for 15 people to sit around the table so that no two arrangements have the same neighbours.

Answer 1

15 people can sit around a table in (15 – 1)! = 14! ways.
Total number of arrangements = 14!
Now, the number of arrangements in which any person can have the same neighbours on either side by clockwise or anticlockwise arrangements = `(14!)/(2!)`
∴ The number of arrangements in which no two arrangements have the same neighbours

= `14! - (14!)/(2!)`

= `14!(1 - 1/2)`

= `14! xx 1/2`

= `(14!)/(2!)`.