Step 1: Determine the oxidation state of chromium.
The complex is
\[
[Cr(NH_3)_3(H_2O)_2Cl]Cl_2
\]
The two chloride ions outside the coordination sphere contribute
\[
-2
\]
charge.
Hence, the complex ion has charge
\[
+2
\]
Let oxidation state of chromium be \(x\).
Since \(NH_3\) and \(H_2O\) are neutral ligands,
\[
x-1=+2
\]
\[
x=+3
\]
Therefore, chromium is in the \(+3\) oxidation state.
Step 2: Arrange ligand names alphabetically.
Ligands present are:
\[
NH_3 \rightarrow ammine
\]
\[
H_2O \rightarrow aqua
\]
\[
Cl^- \rightarrow chlorido
\]
Ignoring multiplicative prefixes, alphabetical order is
\[
ammine,\ aqua,\ chlorido
\]
Step 3: Write ligand names with prefixes.
Three ammine ligands:
\[
triammine
\]
Two aqua ligands:
\[
diaqua
\]
One chlorido ligand:
\[
chlorido
\]
Thus,
\[
\text{Triamminediaqua chlorido chromium (III)}
\]
Since chloride is the counter ion outside the coordination sphere, add
\[
\text{chloride}
\]
at the end.
Step 4: Final conclusion.
Hence, the IUPAC name is
\[
\boxed{\text{Triamminediaqua chlorido chromium (III) chloride}}
\]
Therefore, the correct option is (1).