12. Let $S$ be a finite set, $|S|=n, k$ is a positive integer, (1) Find the number of ordered $k$-tuples of subsets $\left(S_{1}, S_{2}, \cdots, S_{k}\right)$ of $S$ such that $\bigcap_{i=1}^{k} S_{i}=\varnothing$; (2) If $S_{1} \subseteq S_{2} \subseteq \cdots \subseteq S_{k}$, find the number of such $k$-tuples of subsets $\left(S_{1}, S_{2}, \cdots, S_{k}\right)$.
12. (1) Correspond the subsets $S_{1}, S_{2}, \cdots, S_{k}$ of $S=\left\{x_{1}, x_{2}, \cdots, x_{n}\right\}$ to a $k \times n$ table of 0s and 1s.

If $x_{j} \in S_{i}$, then fill the cell in the $i$-th row and $j$-th column with 1, otherwise fill it with 0. $\bigcap_{i=1}^{k} S_{i}=\varnothing$ is equivalent to no column in this table being entirely 1.

Let $T$ be the set of all $k \times n$ tables where no column is entirely 1. Then the set of ordered subset groups $\left\{S_{1}, S_{2}, \cdots, S_{k}\right\}$ that meet the requirement corresponds one-to-one with $T$, and it is easy to see that $|T|=\left(2^{k}-1\right)^{n}$.
（2）Using the $k \times n$ table from (1), $S_{1} \subseteq S_{2} \subseteq \cdots \subseteq S_{k}$ is equivalent to each column in the table (from top to bottom) being a non-decreasing sequence of 0s and 1s. There are exactly $k+1$ such sequences, so there are $(k+1)^{n}$ such tables.