7. Given
$$
\begin{array}{l}
A \cup B \cup C=\{a, b, c, d, e\}, A \cap B=\{a, b, c\}, \\
c \in A \cap B \cap C .
\end{array}
$$

Then the number of sets $\{A, B, C\}$ that satisfy the above conditions is. $\qquad$
7. 100 .

As shown in Figure 3, the set $A \cup B \cup C$ can be divided into seven mutually exclusive regions, denoted as $1, 2, \cdots, 7$.

It is known that element $c$ is in region 7, elements $a, b$ can appear in each of the regions 4, 7, with 4 possibilities; elements $d, e$ can appear in each of the regions $1, 2, 3, 5, 6$, with 5 possibilities each.

Thus, the number of sets $\{A, B, C\}$ that meet the conditions is $4 \times 5 \times 5=$ 100.