Example 5 (1990 Canadian Mathematical Olympiad) Arrange $\frac{n(n+1)}{2}$ different numbers randomly into a triangular array as shown in Figure $12-1$. Let $M_{k}$ be the maximum number in the $k$-th row (from top to bottom). Try to find the probability that $M_{1}<M_{2}<\cdots<M_{n}$.
Let the required probability be $p_{n}$, then $p_{1}=1$.
For a table with $n+1$ rows, the largest number $\frac{(n+1)(n+2)}{2}$ should be in the $n+1$-th row, the probability of which is
$$
\frac{n+1}{\frac{(n+1)(n+2)}{2}}=\frac{2}{n+2} \text {. }
$$

Then the $(n+1)$-th row can be filled arbitrarily, and finally, the remaining $\frac{n(n+1)}{2}$ numbers are filled into the first $n$ rows. The probability that meets the requirement is $p_{n}$, so $\quad p_{n+1}=\frac{2}{n+2} p_{n}$.
Therefore, $p_{n}=\frac{2}{n+1} \cdot \frac{2}{n} \cdots \cdot \frac{2}{3} p_{1}$
$$
=\frac{2^{n}}{(n+1)!} .
$$