Permutation & Combination • Topic 3 of 4

Circular Arrangements

Around a circle there is no fixed "first" seat, so a rotation of an arrangement is the same arrangement. Fixing one person to kill the rotational symmetry leaves (n−1)! ways to seat n distinct people at a round table. If the arrangement can also be flipped over and still count as the same — a necklace of beads or a garland of flowers, where clockwise and anticlockwise are indistinguishable — divide once more by 2, giving (n−1)!/2. The CAT subtlety: a round table where seats are numbered or where there is a head (a throne, a fixed gate) is NOT a free rotation, so it behaves like a row and you use n! instead. Always ask two questions: can it rotate freely (÷ by removing one factor), and can it be reflected (÷ 2)?

✅ Solved examples

1. In how many ways can 5 people sit around a round table?
Circular: (5 − 1)! = 4! = 24.
2. How many distinct necklaces can be made with 6 different beads?
Necklace (rotations and reflections same): (6 − 1)!/2 = 120/2 = 60.
3. 7 people sit around a round table; 2 particular friends must sit together. How many ways?
Treat the pair as one unit ⇒ 6 units around a circle = (6 − 1)! = 120, and the pair can swap internally in 2! = 2. Total = 120 × 2 = 240.
4. In how many ways can 4 couples sit around a round table so each couple sits together?
Glue each couple into a block ⇒ 4 blocks around a circle = (4 − 1)! = 6; each block arranges internally in 2! = 2, four blocks give 2⁴ = 16. Total = 6 × 16 = 96.

✏️ Practice — try these, take hints as needed

1. In how many ways can 6 people sit around a round table?
Use (n − 1)!.
That is 5!.
5 × 4 × 3 × 2 × 1.
120
2. How many garlands can be made from 5 distinct flowers?
Garland = necklace.
(n − 1)!/2.
4!/2 = 24/2.
12
3. 8 people around a round table with a fixed numbered head seat. How many ways?
Fixed/numbered seats kill rotation symmetry.
It behaves like a row.
Use 8!.
40,320
4. 6 people around a round table; 2 particular people must sit together. How many ways?
Tie the pair into one unit.
5 units circular = (5 − 1)!.
Multiply by 2! for the pair.
48
5. In how many ways can 5 keys be put on a circular key ring?
Key ring can be flipped.
So it is a necklace.
(5 − 1)!/2.
12

📝 Topic test — 8 questions

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