Seven people seated round a table in age order

Seven people, no two of the same age, take seats around a circular table in a uniformly random arrangement. What is the probability that going around the table their ages are in sorted order, whether increasing clockwise or counter-clockwise?

$1/360$. Up to rotation there are $6!=720$ equally likely circular orders. Exactly two of them (ascending one way or the other) are sorted, giving $\tfrac{2}{720}$.