Worked Solution
How to Think About It: Around a round table, we fix one person's seat to eliminate rotational equivalence. The remaining 6 people can sit in $6!$ equally likely arrangements. For strict age order, once the direction (clockwise or counterclockwise) is chosen, the seating is completely determined -- there is exactly one clockwise ordering and one counterclockwise ordering. So there are 2 valid arrangements out of $6! = 720$.
Quick Estimate:
/720 = 1/360 \approx 0.0028$. About a quarter of one percent.
Approach: Count circular permutations and favorable outcomes.
Formal Solution:
For $n$ people around a circular table, the number of distinct seating arrangements is $(n-1)!$, since we fix one person to break rotational symmetry.
With $n = 7$: total arrangements $= 6! = 720$.
For strict age order around the circle, we need the ages to be monotonically increasing (or decreasing) as we go around. Fix person 1 (say, the youngest) in a seat. Then: - Clockwise order: The remaining 6 people must sit in exactly one specific arrangement (ages increasing clockwise). That is 1 arrangement. - Counterclockwise order: The remaining 6 must sit in the reverse order. That is 1 arrangement.
So there are exactly 2 favorable outcomes.
$P(\text{age-ordered seating}) = \frac{2}{(7-1)!} = \frac{2}{720} = \frac{1}{360}$
Answer: $P = \dfrac{2}{720} = \dfrac{1}{360}$.
Intuition
Circular permutation problems always start the same way: fix one person to break rotational symmetry, then count the remaining arrangements. The strict age ordering constraint is extremely restrictive -- out of $6! = 720$ possible seatings, only 2 work (one for each direction). This is because once you fix the starting point and choose a direction, every seat is determined.
The general formula for $n$ people is
/(n-1)!$. Notice how fast this drops: for 4 people it is
/6 = 1/3$, for 7 it is
/360$, for 10 it is
/9! \approx 5.5 \times 10^{-6}$. This exponential decay in the probability of ordered configurations is related to the concept of entropy -- random arrangements are overwhelmingly likely to be disordered. The same idea underlies why portfolios of many assets almost never have all positive (or all negative) returns on the same day, and why order statistics spread out predictably.