Example 1.2 Let $n$ be a positive integer greater than 1, find the number of ordered pairs of positive integers $(x, y)$ that satisfy the condition $x+y \leqslant n$.
Solution: Let the required number be $N$. Since the number of ordered pairs of positive integers $(x, y)$ that satisfy the condition $x+y \leqslant n$ and $x=k(1 \leqslant k \leqslant n-1)$ is $n-k$, by the principle of addition, we have
$$
\begin{aligned}
N & =\sum_{k=1}^{n-1}(n-k) \\
& =(n-1)+(n-2)+\cdots+2+1 \\
& =\frac{n(n-1)}{2} .
\end{aligned}
$$