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Question

Difficulty: MediumPermutations

A committee of 66 delegates is to be seated around a circular table for a conference. If two specific delegates must always sit next to each other, in how many distinct ways can the delegates be arranged?

  1. 48Answer
  2. B
    24
  3. C
    240
  4. D
    15

Answer

48
The correct answer is 48 because treating the 2 restricted delegates as a single unit leaves 5 items to arrange around a circle, which yields (5−1)!=24(5-1)! = 24 arrangements. Since the 2 delegates can arrange themselves in 2!=22! = 2 ways within their block, the total number of arrangements is 24×2=4824 \times 2 = 48.

Step-by-Step Solution

1
Group the restricted delegates into a single block
2 specified delegates are treated as 1 unit, leaving 4 remaining delegates, making a total of 5 items to arrange.
Since the two delegates must sit next to each other, treating them as a single entity ensures they remain together in all seating positions.
2
Calculate the circular arrangements of the 5 items
(5−1)!=4!=24(5 - 1)! = 4! = 24 ways.
The number of ways to arrange nn distinct items around a circular table is given by (n−1)!(n - 1)!.
3
Account for internal arrangements of the paired delegates
2!=22! = 2 ways.
The two delegates within the single block can swap positions between themselves.
4
Multiply the circular arrangements by the internal arrangements
24×2=4824 \times 2 = 48 total distinct arrangements.
By the fundamental counting principle, total arrangements equal the product of the independent steps.

Key Concept

Circular Permutations with Restrictions
Estimated Time:1m 15s
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Topics

Permutations

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