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

Sum

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!).

Concept: Permutations - Circular Permutations
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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board
Chapter 6 Permutations and Combinations
Exercise 6.5 | Q 4 | Page 85