Circular Block Method
Fix one seat to remove rotations; glue items that must sit together.
What is it?
Circular arrangements of distinct items number . Items that must be adjacent are merged into a block that is arranged internally.
Why is it useful?
Round-table seating and necklace-style questions are standard counting warm-ups.
When should you think of it?
Use it when seating is around a table or items that must be together appear.
Think of this method when…
- Round table, circle, ring
- Rotations are considered the same
- "A and B must sit together"
- Reflections may also be identified (necklaces)
Core idea
Fix one person to kill rotational symmetry; a block of counts as one unit with internal orders.
Step-by-step playbook
- 1Merge any "must be together" items into blocks.
- 2Count units and arrange them in ways.
- 3Multiply by internal arrangements of each block.
- 4Divide by 2 if reflections are identical.
Worked example
Worked example
6 people sit at random around a round table. What is the probability that A and B sit next to each other?
- 1
Total: .
- 2
Glue A and B: 5 units give arrangements, times internal orders.
- 3
Divide.
- Answer:
Common mistakes
- Using for a circle.
- Forgetting internal orders of the block.
- Dividing by 2 when reflections are distinct (chairs at a table).
Would you use this method?
8 people stand in a line. Should you fix one person's position?
Practice this method
9 QuantSolve questions are tagged with this method.
Practice this method