Combinatorics path · C07

Before this lesson: Permutations and arrangements

Your goal: Specify which configurations count as identical before dividing.

Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.

Names you may know: circular arrangements; rotational symmetry.

Circular permutations: the key idea

For n distinct people around an unlabelled round table, rotations represent the same seating, so fix one person and arrange the other n−1: (n−1)! for n≥1. Clockwise and anticlockwise orders are usually different for seating. If mirror images are also identified, as in an unlabelled bracelet model with distinct beads, divide by 2 for n≥3. Repeated colours may have rotational symmetries, so blindly dividing a linear count by n can fail; use a symmetry argument appropriate to that model.

A worked example

How many circular seatings of five distinct people are there when rotations are the same but reflections differ?

Fix one person as reference and arrange the other four clockwise. The count is 4!=24.

Your turn: change one thing

How many bracelets with five distinct beads if rotations and reflections are the same?

Try this on paper before opening the explanation.

Compare your reasoning

There are 24 circular orders before reflection. No order of five distinct beads is fixed by a reflection, so the mirror pairs give 24/2=12 bracelets.

Pause and check

A trap to avoid: Dividing by two for reflections when reflections remain distinct or fixed configurations exist.

Practise and adjust the level

Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.

Interactive practice loads here. You can also use the complete question set below.

Write a complete argument

Explain why fixing one person counts circular seatings exactly once.

Planning hint

List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.

Read the full solution after your attempt

Every circular seating can be rotated to put the chosen person at the reference position. Because that person is unique, this rotation is unique. The remaining clockwise order then determines the seating, giving a bijection with permutations of the other n−1 people.

My proof notebook

Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.

All nine practice questions

Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.

1. In the usual round-table model, which arrangements are identified?

Foundation

  1. Only adjacent swaps
  2. Every reflection automatically
  3. Rotations
  4. All permutations
Hint

Specify the equivalence convention.

Answer and reasoning

Rotations. Moving everyone by the same rotation does not change the circular seating.

2. For n distinct people, the circular seating count is what?

Foundation

  1. n!
  2. n²
  3. 2n2^{n}
  4. (n−1)!
Hint

Fix one reference person.

Answer and reasoning

(n−1)!. The other n−1 people can be ordered freely around the table.

3. Five distinct people around a round table: how many seatings?

Core

  1. 120
  2. 60
  3. 12
  4. 24
Hint

Use 4!.

Answer and reasoning

24. Fix one person and arrange four: 24.

4. Five distinct beads with rotations and reflections identified: count?

Core

  1. 120
  2. 10
  3. 12
  4. 24
Hint

Pair the 24 circular orders with their mirrors.

Answer and reasoning

12. Each reflection pair has size two, giving 12.

5. Why can dividing by n fail with repeated colours?

Stretch

  1. n is never the right divisor
  2. Colours are numbers
  3. Every arrangement has n distinct rotations
  4. Some arrangements have smaller rotation orbits
Hint

A periodic pattern can repeat under rotation.

Answer and reasoning

Some arrangements have smaller rotation orbits. Equal-size rotation classes are needed for simple division; repeated patterns can have nontrivial symmetry.

6. For circular seating, should mirror images be identified without being told?

Stretch

  1. Only for odd n
  2. Only for students
  3. No, state the convention
  4. Yes always
Hint

Different problems use different equivalences.

Answer and reasoning

No, state the convention. Usual seating keeps clockwise and anticlockwise orders distinct.

7. Fixing one person at a round table removes which duplication?

Foundation

  1. All adjacency choices
  2. Rotation
  3. Reflection automatically
  4. Repeated names
Hint

It sets a reference without changing relative order.

Answer and reasoning

Rotation. Each rotation class has exactly one such representative.

8. Four distinct people around a round table, rotations equal: count?

Core

  1. 12
  2. 4
  3. 6
  4. 24
Hint

Use (4−1)!.

Answer and reasoning

6. 3!=6.

9. For distinct beads with n≥3, why can reflection classes be paired?

Stretch

  1. A circular order and its reverse are distinct before identification
  2. Every order equals its reverse
  3. There are no rotations
  4. All bead colours repeat
Hint

Distinct labels rule out reflection symmetry of a circular order.

Answer and reasoning

A circular order and its reverse are distinct before identification. The reflection action has pairs of size two in this distinct-bead setting.

Choose your next step

Continue to Dividing objects into fixed-size groups. If this felt difficult, return to a prerequisite above. Every lesson stays open.

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