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