8. Given that the number of elements in sets $A$, $B$, and $A \cup B$ are $12$, $13$, and $15$ respectively, set $C$ satisfies the following three conditions: (1) $C \subseteq A \cup B$; (2) The number of elements in $C \cap A \cap B$ is 1; (3) $C \nsubseteq B$. How many such sets $C$ are there? $\qquad$
8. 240 .

Let $I=A \cup B$, then the number of elements in $A \cap B$, $A \cap \bar{B}$, and $\bar{A} \cap B$ are 10, 2, and 3, respectively. The number of sets $C$ that satisfy conditions (1) and (2) is $C_{10}^{1} \cdot 2^{5}=320$ (sets), among which $C_{10}^{1} \cdot 2^{0} \cdot 2^{3}=80$ (sets) do not satisfy condition (3). Therefore, the number of sets $C$ is $320-80=240$ (sets).