Example 5 Let the set $M=\{1,2,3, \cdots, 1000\}$. For any non-empty subset $X$ of $M$, let $\alpha_{X}$ denote the sum of the largest and smallest numbers in $X$. Then the arithmetic mean of all such $\alpha_{X}$ is $\qquad$
(1991, National High School Mathematics Competition)
Solution: Construct the subset $X^{\prime}=\{1001-x \mid x \in X\}$, then all non-empty subsets can be divided into two categories: $X^{\prime}=X$ and $X^{\prime} \neq X$.

When $X^{\prime}=X$, it must be that $X^{\prime}=X=M$, thus, $\alpha_{X}=1001$.

When $X^{\prime} \neq X$, let $x$ and $y$ be the maximum and minimum numbers in $X$, respectively, then $1001-x$ and $1001-y$ are the minimum and maximum numbers in $X^{\prime}$, respectively. Therefore, $a_{X}=x+y$, $\alpha_{X^{\prime}}=2002-x-y$. Thus, $\frac{a_{X}}{2} \frac{\alpha_{X^{\prime}}}{2}=1001$.
Hence, the arithmetic mean of the required $\alpha_{x}$ is 1001.
4 Solution using the principle of inclusion-exclusion
Using the principle of inclusion-exclusion to solve certain subset competition problems can directly calculate the result of the problem.