(10) (15 points) Let $M$ be an $n$-element set. If $M$ has $k$ distinct subsets $A_{1}, A_{2}, \cdots, A_{k}$, satisfying: for each $i, j \in\{1,2, \cdots, k\}, A_{i} \cap A_{j} \neq \varnothing$, find the maximum value of the positive integer $k$.


10 The maximum value of a positive integer $k$ is $2^{n-1}$.
(1) First prove: there exist $2^{n-1}$ subsets of $M$,

such that any two of them have a non-empty intersection.
Let $M=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$, and $A_{1}, A_{2}, \cdots, A_{2^{n-1}}$ be all $2^{n-1}$ subsets of $\left\{a_{1}, a_{2}, \cdots, a_{n-1}\right\}$. Let $B_{i}=A_{i} \cup\left\{a_{n}\right\}, i=1,2, \cdots, 2^{n-1}$, then the $2^{n-1}$ subsets $B_{1}, B_{2}, \cdots, B_{2^{n-1}}$ of $M$ have a non-empty intersection with each other.
(2) Then prove: for any $2^{n-1}+1$ subsets of $M$, there must be two subsets that are disjoint.

Let $B_{1}, B_{2}, \cdots, B_{2^{n-1}}$ be $2^{n-1}$ different subsets of $M$, each containing $a_{n}$; let $\bar{B}_{i}$ denote the complement of subset $B_{i}$ in $M$, $\bar{B}_{i}=M \backslash B_{i}, i=1,2, \cdots, 2^{n-1}$, then for any $i \neq j, B_{i} \neq B_{j}, \bar{B}_{i} \neq \bar{B}_{j}$, and $B_{i} \neq \bar{B}_{j}$ (since the former contains $a_{n}$ and the latter does not), hence $B_{1}, B_{2}, \cdots, B_{2^{n-1}}, \bar{B}_{1}, \bar{B}_{2}, \cdots, \bar{B}_{2^{n-1}}$ are all $2^{n}$ different subsets of $M$. Now pair these subsets into $2^{n-1}$ pairs: $\left(B_{1}, \bar{B}_{1}\right)$, $\left(B_{2}, \bar{B}_{2}\right), \cdots,\left(B_{2^{n-1}}, \bar{B}_{2^{n-1}}\right)$. Taking any $2^{n-1}+1$ subsets of $M$, there must be two subsets belonging to the same pair, then these two subsets are disjoint.