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  1. LPU MCA Semester 1
  2. PEA515
  3. Unit 5

PEA515 · Unit 5: Permutation, combination and probability

Circular arrangement

30 practice questions · in the ETE syllabus

Practise this topicPractise this topic

Key points

  • Round table: (n - 1)! because rotations are the same
  • Pair together: 2 x (n - 2)!
  • Alternating groups: seat one group, then fill gaps
  • Necklace of n beads: (n - 1)!/2

Around a circular table only the relative positions matter, so rotating everyone gives the same seating. Fix one person to remove the rotations: n people can sit around a table in (n - 1)! ways.

ProblemCount
n people around a table(n - 1)!
Two named people together2 x (n - 2)!
Two named people not together(n - 1)! - 2 x (n - 2)!
n boys and n girls alternating(n - 1)! x n! (seat one group, then fill the gaps)
n different beads on a necklace or garland(n - 1)! / 2 (it can also be turned over)

Example: 6 people with A and B together: glue them into a block, so there are 5 objects, giving (5 - 1)! = 24 arrangements, doubled for the order of A and B, so 48. For 3 boys and 3 girls alternating: (3 - 1)! x 3! = 12.

In a row the formulas change (n! and 2 x (n - 1)! for a pair together), because a row has a fixed start.

Exam trap: a necklace or bracelet can be flipped over, so divide by 2 as well, but a seating at a round table cannot be flipped.

Try two questions

Question 1 · Tier 3, Apply

In how many different ways can 4 people be seated around a circular table?

  1. 24
  2. 2
  3. 6
  4. 12
Show the answer

C. Fix one person to remove rotations and arrange the other 3: (4 - 1)! = 6.

Why not A: Counts the rotations of the same seating as different.

Why not B: Fixes two people instead of one.

Why not D: Applies a permutation of two items.

Question 2 · Tier 2, Understand

In how many ways can 7 people sit around a circular table if two particular persons A and B must sit next to each other?

  1. 720
  2. 120
  3. 1440
  4. 240
Show the answer

D. Treat A and B as one block: (7 - 1 - 1)! = 120 arrangements, times 2 for their order = 240.

Why not A: Ignores the condition.

Why not B: Forgets that the two can swap places.

Why not C: Uses the row formula for the block.