MathematicsChapter 6: Permutation and Combination

Circular Permutations

Arrangements around a circle: rotating the whole table is the same seating in the school model.

Learning objectives

State the main ideas and notation used in circular permutations.
Apply the rules of circular permutations in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

Arrangements around a circle: rotating the whole table is the same seating in the school model. n different people around a round table: (n-1)!. If clockwise and anticlockwise are the same (necklace, some bracelet questions), divide by 2 as the book says: (n-1)!/2. Meaning: fix one person's seat to kill the rotation, then arrange the others in the remaining n-1 seats.

Where / symbols

CIRCULAR PERMUTATIONS: Arrangements around a circle: rotating the whole table is the same seating in the school model. n different people around a round table: (n-1)!. If clockwise and anticlockwise are the same (necklace, some bracelet questions), divide by 2 as the book says: (n-1)!/2. Meaning: fix one person's seat to kill the rotation, then arrange the others in the remaining n-1 seats.
1. ROUND TABLE: 5 people: 4! = 24. One person already sitting: then 4! leftover, which matches (5-1)!. A head of table named in a wedding question may make it linear again — read carefully.
2. FLIPS: If the arrangement can be flipped and counted as the same, (n-1)!/2. Beads on a necklace often. People at a dinner usually have a left and right, so do not divide by 2 unless the book says so.

Overview

Fix one seat, then the necklace:

Round Table

5 people: 4! = 24. One person already sitting: then 4! leftover, which matches (5-1)!. A head of table named in a wedding question may make it linear again — read carefully.

Examples (n-1)! · Not n! · Relative seats matter

Flips

If the arrangement can be flipped and counted as the same, (n-1)!/2. Beads on a necklace often. People at a dinner usually have a left and right, so do not divide by 2 unless the book says so.

Examples Dinner: (n-1)! · Necklace: maybe /2 · Copy the book's convention

WHAT — CIRCULAR PERMUTATIONS

RULE — 1. ROUND TABLE

5 people: 4! = 24. One person already sitting: then 4! leftover, which matches (5-1)!. A head of table named in a wedding question may make it linear again — read carefully.

METHOD — 2. FLIPS

If the arrangement can be flipped and counted as the same, (n-1)!/2. Beads on a necklace often. People at a dinner usually have a left and right, so do not divide by 2 unless the book says so.

WORKED EXAMPLE

Worked: 6 distinct people round a table (no flip). (6-1)! = 5! 5! = 120 not 720 (that would be a line)

How to answer a circular permutations question

Identify the given quantities, notation, and required result.
Choose the matching rule and write it before substitution.
Work one algebraic step at a time, keeping brackets and signs clear.
Check restrictions, simplify the result, and verify it where possible.

Worked example

6 distinct people round a table (no flip).
(6-1)! = 5!
5! = 120
not 720 (that would be a line)

In simple words

A line is n!. A plain round table is (n-1)!. Do not mix them.

Key points

round table: (n-1)!
necklace often /2
fix one person
left-right or not

Exam preparation

Exam Focus

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Review the key facts, formulas and checks before you move on.