Six people sit randomly around a round table. What is the probability that they are seated in increasing order by their age?

Free worked solution · Quant Blueprint

1/120 for a specified direction around the table, assuming distinct ages.

1. A circle has no first seat. Interpret increasing order as starting at the youngest and proceeding clockwise, with the wrap back to the youngest allowed.
2. Fix the youngest person in one seat to remove rotational duplicates. The other five people have 5! equally likely arrangements.
3. Exactly one arrangement places the remaining people in increasing age order, giving 1/5! = 1/120.

Common mistake

A literal strict increase all the way around a closed circle is impossible. Clarify the starting person, direction and whether ages are distinct.

Follow-up

If either clockwise or counterclockwise increasing order is accepted, there are two arrangements and the probability is 2/120 = 1/60.

Original educational explanation for this practice question. Reviewed 2026-10-13. Firm-style practice; the label describes the collection’s preparation context.