5. From the set $\{1,2, \cdots, 10\}$, any two non-adjacent numbers are taken and multiplied. Then the sum of all such products is equal to


5. 990 .

Take any two numbers, multiply them, and then find their sum:
$$
\begin{array}{l}
S_{1}=\frac{1}{2} \sum_{k=1}^{10} k\left(\sum_{i=1}^{10} i-k\right) \\
=\frac{1}{2} \sum_{k=1}^{10} k(55-k)=\frac{1}{2} \sum_{k=1}^{10}\left(55 k-k^{2}\right) \\
=\frac{1}{2}(55 \times 55-385)=1320,
\end{array}
$$

Among them, the sum that does not meet the condition is
$$
\begin{array}{l}
S_{2}=\sum_{k=1}^{9} k(k+1)=\sum_{k=1}^{2}\left(k^{2}+k\right) \\
=285+45=330 .
\end{array}
$$

Therefore, the required sum is
$$
S=S_{1}-S_{2}=1320-330=990 .
$$