8. Let $S$ be a non-empty subset of the set $\{1,2, \cdots, 15\}$. If a positive integer $n$ satisfies: $n \in S, n+|S| \in S$, then $n$ is called a "model number" of the subset $S$, where $|S|$ denotes the number of elements in the set $S$. For all non-empty subsets $S$ of the set $\{1,2, \cdots, 15\}$, the sum of the number of model numbers is $\qquad$
$8.13 \times 2^{12}$. 
Just find, for each $n$, how many subsets there are such that $n$ is a model number, and then sum over $\mathrm{F} n$.

If $n$ is a model number of $S$, and $S$ contains $k$ elements, then $n \in S, n+k \in S$. Therefore, $k \geqslant 2$.

Also, $k \leqslant 15-n$, and the other $k-2$ elements of $S$ can be chosen in $\mathrm{C}_{13}^{k-2}$ ways. Hence, when $n$ is a model number, there are
$$
\mathrm{C}_{13}^{0}+\mathrm{C}_{13}^{1}+\cdots+\mathrm{C}_{13}^{13-n} \text { (ways). }
$$

When $n=1,2, \cdots, 13$, the total number of model numbers is
$$
A=13 \mathrm{C}_{13}^{0}+12 \mathrm{C}_{13}^{1}+\cdots+\mathrm{C}_{1,3}^{12} \text {. }
$$

Therefore, $2 A=13 \mathrm{C}_{13}^{0}+12 \mathrm{C}_{13}^{1}+11 \mathrm{C}_{13}^{2}+\cdots+\mathrm{C}_{13}^{12}+$
$$
\begin{aligned}
& \quad \mathrm{C}_{13}^{1}+2 \mathrm{C}_{13}^{2}+\cdots+12 \mathrm{C}_{13}^{12}+13 \mathrm{C}_{13}^{13} \\
= & 13\left(\mathrm{C}_{13}^{0}+\mathrm{C}_{13}^{1}+\mathrm{C}_{13}^{2}+\cdots+\mathrm{C}_{13}^{12}+\mathrm{C}_{13}^{13}\right) \\
= & 13 \times 2^{13} .
\end{aligned}
$$

Thus, for all non-empty subsets $S$ of the set $\{1,2, \cdots, 15\}$, the sum of the number of model numbers is $13 \times 2^{12}$.