Question 30
A set of $\frac{1}{2} n(n+1)$ distinct numbers is arranged at random in a triangular array:
Let $M_{k}$ be the largest number in the $k$-th row from the top. Find the probability that
$$
M_{1}\lt M_{2}\lt M_{3}\lt \cdots\lt M_{n}
$$
Let $M_{k}$ be the largest number in the $k$-th row from the top. Find the probability that
$$
M_{1}\lt M_{2}\lt M_{3}\lt \cdots\lt M_{n}
$$