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Question

Difficulty: MediumPermutations

Six members of a board of directors are to be seated around a circular conference table. If two specific members refuse to sit next to each other, how many different seating arrangements are possible?

  1. 72Answer
  2. B
    48
  3. C
    120
  4. D
    240

Answer

72
To find the number of circular seating arrangements where two specific members do not sit next to each other, use the complement rule. First, the total unrestricted circular arrangements of 6 members is (6−1)!=5!=120(6 - 1)! = 5! = 120. Next, calculate the arrangements where the two members sit together by treating them as 1 unit (giving 5 units in total). The circular arrangements of these 5 units is (5−1)!=4!=24(5 - 1)! = 4! = 24, and the 2 members can swap seats in 2!=22! = 2 ways, yielding 24×2=4824 \times 2 = 48 arrangements together. Subtracting this from the total yields 120−48=72120 - 48 = 72.

Step-by-Step Solution

1
Calculate total circular arrangements without restrictions
Total arrangements = (6−1)!=5!=120(6 - 1)! = 5! = 120
The number of ways to arrange nn distinct items in a circle is (n−1)!(n - 1)!.
2
Calculate arrangements where the two specific members sit together
Restricted arrangements = (5−1)!×2!=4!×2=24×2=48(5 - 1)! \times 2! = 4! \times 2 = 24 \times 2 = 48
Treat the two members as a single block, giving 5 units to arrange around a circle in (5−1)!(5-1)! ways, and multiply by 2!2! for internal ordering of the pair.
3
Subtract the together arrangements from total arrangements
Ways apart = 120−48=72120 - 48 = 72
Complementary counting gives the number of arrangements where the two members do not sit next to each other.

Key Concept

Circular Permutations with Restrictions
Estimated Time:1m 30s
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Topics

Permutations

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