6. Choose any two non-adjacent numbers from $1,2, \cdots, 10$ and multiply them. The sum of all such products is $\qquad$
6.990.

The sum of the products of any two different numbers taken is
$$
\frac{1}{2} \sum_{k=1}^{10} k\left(\sum_{i=1}^{10} i-k\right) \text {. }
$$

The sum of the products of any two adjacent numbers is $\sum_{k=1}^{9} k(k+1)$.
The sum that satisfies the condition is
$$
\begin{array}{l} 
\frac{1}{2} \sum_{k=1}^{10} k\left(\sum_{i=1}^{10} i-k\right)-\sum_{k=1}^{9} k(k+1) \\
= \frac{1}{2} \sum_{k=1}^{9}[k(55-k)-2 k(k+1)] \\
+\frac{1}{2} \times 10 \times(55-10) \\
= \frac{1}{2} \sum_{k=1}^{9} k(53-3 k)+225 \\
= \frac{53}{2} \sum_{k=1}^{9} k-\frac{3}{2} \sum_{k=1}^{9} k^{2}+225 \\
= \frac{53 \times 45}{2}-\frac{45 \times 19}{2}+225=990
\end{array}
$$