Stand Up & Math · Grade 9 · Activity
Round Table Trouble
Circular permutations · Arrangements vs selections · Groups of 3–4 · Whiteboard
You're planning the seating for a major round-table event. The client keeps adding rules, and every miscount means the wrong number of place settings gets ordered. Precision is everything — and this time, a number alone won't be enough. You'll need to convince the client your formula is always right, not just right for the case you happened to check.
How many ways can 3 people, A, B and C, be arranged in a straight line? Write out the full list.
Now imagine those same 3 people seated around a circular table instead. Go through your list from Task 1a — for each line-up, note who's sitting immediately to each person's right once they're wrapped into a circle. Do any of your 6 line-ups end up describing the exact same circular seating as another?
Based on what you found, how many genuinely different circular seatings are there for 3 people? Try a few other values of \(n\) and find a general formula for \(n\) people around a circular table.
Task 1d — Your formula should work for any number of people \(n\), not just the small cases you checked by hand. Write a convincing argument for your group's whiteboard explaining why the formula is always correct — not just that it happens to match the examples you tried.
You've worked through all four phases of Round Table Trouble.
Well done — get ready for the class consolidation.