Seven people around a table sorted by age

Seven people, all of different ages, take the seven seats at a round table in a uniformly random arrangement. What is the probability that going around the table, in one direction or the other, their ages increase steadily from the youngest to the oldest?

$\tfrac1{360}$. Fix the youngest person's seat; the other six are then in one of $6!=720$ equally likely orders. Exactly two of these (ages increasing clockwise or anticlockwise) qualify, so the probability is $\tfrac{2}{720}=\tfrac1{360}$.