Step 1: Split the formula.
The compound $[\text{Co}(\text{H}_2\text{O})(\text{NH}_3)_5]\text{I}_3$ has a complex cation inside the brackets and three iodide ions sitting outside as counter-ions.
Step 2: Find the cobalt oxidation state.
Water and ammonia are neutral ligands, while the three outer iodides total $-3$. For overall neutrality the cation must be $+3$, so cobalt is $+3$.
Step 3: Name each ligand.
$\text{NH}_3$ becomes ammine and $\text{H}_2\text{O}$ becomes aqua under current IUPAC rules.
Step 4: Apply numerical prefixes.
Five ammine ligands give pentaammine and one aqua ligand stays aqua.
Step 5: Order them alphabetically.
Comparing the ligand names, ammine precedes aqua, so we write pentaammineaqua before the metal.
Step 6: Add the metal and counter-ion.
Since the complex is a cation we keep the plain metal name with its oxidation state, giving cobalt(III), and finish with the anion iodide.
Step 7: Assemble the full name.
Combining everything yields Pentaammineaquacobalt (III) iodide, which is option (1).
\[ \boxed{\text{Pentaammineaquacobalt (III) iodide}} \]