Step 1: Understand the naming convention for coordination compounds.
The IUPAC name of a coordination compound follows the format:
\[
\text{Ligand names} + \text{Metal name} + \text{Oxidation state of metal} + \text{Counterion}.
\]
For the complex [Co(NH$_3$)$_6$]Cl$_3$, the ligands are NH$_3$ (ammine), cobalt (Co) is the metal, and the oxidation state of cobalt is +3 (since the complex is neutral and the chloride ions are -1 each).
Step 2: Apply the IUPAC naming conventions.
- The complex contains 6 NH$_3$ ligands, which are called "ammine".
- The oxidation state of cobalt is +3, so it is named "cobalt (III)".
- The counterion is chloride (Cl), which is written as "chloride" at the end.
Step 3: Apply to the options.
The correct IUPAC name for the compound [Co(NH$_3$)$_6$]Cl$_3$ is "Hexa-ammine cobalt (III) chloride".
Final Answer:
\[
\boxed{\text{Hexa-ammine cobalt (III) chloride}}
\]