Step 1: Define an asymmetric carbon.
An asymmetric carbon atom (chiral center) is a carbon atom bonded to four different groups.
Step 2: Examine the structure of the given molecule: HOCH$_2$-CH(Br)-CH(Br)-CH$_2$OH.
Let's analyze each carbon atom in the chain:
\begin{itemize}
\item Carbon 1 (HOCH$_2$-): Bonded to -OH, -H, -H, and -CH(Br)CH(Br)CH$_2$OH. It has two identical hydrogen atoms, so it is not asymmetric.
\item Carbon 2 (-CH(Br)-): Bonded to -H, -Br, -CH$_2$OH, and -CH(Br)CH$_2$OH. The four groups are different. Therefore, Carbon 2 is asymmetric.
\item Carbon 3 (-CH(Br)-): Bonded to -H, -Br, -CH(Br)CH$_2$OH, and -CH$_2$OH. The four groups are different. Therefore, Carbon 3 is asymmetric.
\item Carbon 4 (-CH$_2$OH): Bonded to -OH, -H, -H, and -CH(Br)CH(Br)CH$_2$OH. It has two identical hydrogen atoms, so it is not asymmetric.
\end{itemize}
Step 3: Count the number of asymmetric carbons.
There are two asymmetric carbon atoms (Carbon 2 and Carbon 3) in the given molecule.
Final Answer: The final answer is $\boxed{2}$