Example 1 Let $X_{n}=\{1,2, \cdots, n\}$, for any non-empty subset $A$ of $X_{n}$, let $T(A)$ be the product of all numbers in $A$. Find $\sum_{A \subseteq X_{n}} T(A)$.
Explanation: Constructive Function
$$
\begin{array}{l} 
f(x)= \sum_{1 \leq X_{n}} T(A) x^{n-|A|} \\
=(1+2+\cdots+n) x^{n-1}+x^{n-2} \sum_{1 \leqslant i<j \leqslant n} \ddot{j}+ \\
\quad x^{n-3} \sum_{1 \leqslant i<j<k \leqslant n} \ddot{j} k+\cdots+n!x^{0} \\
=(x+1)(x+2) \cdots(x+n)-x^{n},
\end{array}
$$

where $|A|$ denotes the number of elements in $A$.
$$
\begin{array}{l} 
\text { Hence } \sum_{A \subseteq X_{n}} T(A) \\
= 1+2+\cdots+n+\sum_{1 \leqslant i<j \leqslant n} \ddot{j}+ \\
\sum_{1 \leqslant i<j<k \leqslant n} \ddot{j} k+\cdots+n! \\
= f(1)=(1+1)(1+2) \cdots(1+n)-1 \\
=(n+1)!-1 .
\end{array}
$$