1. For each pair of distinct numbers in the set $\{1,2, \cdots, 19\}$, the sum of all such products is
$$
\begin{array}{l}
\sum_{1 \leqslant i<j \leqslant 19} i j=\frac{1}{2}\left[\left(\sum_{k=1}^{19} k\right)^{2}-\sum_{k=1}^{19} k^{2}\right]=\frac{1}{2}\left[\left(\frac{19 \times 20}{2}\right)^{2}-\frac{19 \times 20 \times 39}{6}\right] \\
=\frac{1}{2}\left(190^{2}-190 \times 13\right)=95 \times 177=16815 .
\end{array}
$$

The translation is as follows:

$$
\begin{array}{l}
\sum_{1 \leqslant i<j \leqslant 19} i j=\frac{1}{2}\left[\left(\sum_{k=1}^{19} k\right)^{2}-\sum_{k=1}^{19} k^{2}\right]=\frac{1}{2}\left[\left(\frac{19 \times 20}{2}\right)^{2}-\frac{19 \times 20 \times 39}{6}\right] \\
=\frac{1}{2}\left(190^{2}-190 \times 13\right)=95 \times 177=16815 .
\end{array}
$$