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Exercise 6.3 · Q17

Q.In how many ways can 9 people be arranged around a circular table if two people insist on sitting next to each other.

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Bind the 2 people who insist on sitting together into a single unit, arrange the resulting units in a circle, then multiply by the internal arrangements of the bound pair.

[!FORMULA] For kk distinct objects around a circle, the number of arrangements =(k−1)!=(k-1)! (one position is fixed to remove rotational symmetry). If 2 particular objects must stay adjacent, tie them into 1 unit first, then multiply by 2!2! for their own internal order.

  1. There are 99 people; the 22 who insist on being together are tied into a single block, so effectively 9−2+1=89-2+1=8 units must be seated around the table.
  2. Circular arrangements of these 88 units =(8−1)!=7!=(8-1)! = 7!.
  3. 7!=50407! = 5040.
  4. Within the tied block, the 2 people can be ordered between themselves in 2!=22! = 2 ways.
  5. Total arrangements =7!×2!=5040×2=10080=7!\times 2! = 5040\times 2 = 10080. …
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