A set of n(n + 1)/2 (distinct) numbers is arranged at random in a triangular array:
*
* *
* * *
.
.
* * ... * *
Let Mk denote the largest number in the kth row from the top. Find the probability that Mn > Mn-1 > ... > M1.
The kth triangle has k(k+1)/2 numbers. In order for the condition to hold, the largest of those k(k+1)/2 numbers must be in the kth row. Since the kth row has k numbers, this happens with probability k/(k(k+1)/2) = 2/(k+1).
This must be true for each of the n triangles.
The desired probability = Product (where k goes from 1 to n) of 2/(k+1).
Desired probability = 2^n / (n+1)!
Solution (spoiler)