2. Let $m, n \in N$, and $m>n$, set $A=\{1$, $2,3, \cdots, m\}, B=\{1,2,3, \cdots, n\}$, and $C \subset A$. Then the number of $C$ that satisfies $B \cap C \neq \varnothing$ is $\qquad$ .
2. $2^{m-n}\left(2^{n}-1\right)$.

Since the subsets of $A$ that consist only of numbers chosen from $n+1, n+2, \cdots, m$ are the only ones that can satisfy $B \cap C = \varnothing$, and there are $2^{m-n}$ such subsets, the number of $C$ that satisfy $B \cap C \neq \varnothing$ is $2^{m}-2^{m-n}=2^{m-n}\left(2^{n}-1\right)$ (sets).