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Question
Fifteen persons sit around a table. Find the number of arrangements that have two specified persons not sitting side by side.
Solution
Here, n = 15
15 persons are arranged around the table in (15 – 1)! = 14! ways.
Let two particular persons sit together.
∴ the number of ways in which 15 persons are arranged around the table such that two particular persons sit together is = 13! × 2!
Hence, the number of ways in which 15 persons are arranged around the table such that two particular persons are not to be side by side
= 14! – 13! × 2!
= 14 × 13! – 13! × 2
= 13!(14 – 2)
= 12 × 13!
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