According to Werner's coordination theory:

1. Coordination Number: Cobalt(III) typically has a coordination number (secondary valency) of 6 in these ammine complexes.
2. Ionisation: Chloride ions present outside the coordination sphere (primary valency only) are ionisable and will precipitate as AgCl with silver nitrate. Chloride ions inside the coordination sphere (secondary valency) are non-ionisable and do not precipitate.
3. Formulation: We arrange the ligands ($NH_3$ and $Cl^-$) to satisfy the coordination number of 6 inside the square brackets: $CoCl_3 \cdot 6NH_3 \rightarrow [Co(NH_3)_6]Cl_3$ (3 ionisable $Cl^-$) $CoCl_3 \cdot 5NH_3 \rightarrow [Co(NH_3)_5Cl]Cl_2$ (2 ionisable $Cl^-$) $CoCl_3 \cdot 4NH_3 \rightarrow [Co(NH_3)_4Cl_2]Cl$ (1 ionisable $Cl^-$) $CoCl_3 \cdot 3NH_3 \rightarrow [Co(NH_3)_3Cl_3]$ (0 ionisable $Cl^-$)

Since $[Co(NH_3)_3Cl_3]$ has no chloride ions outside the coordination sphere, it does not release $Cl^-$ ions into solution and thus gives no precipitate (test) with silver nitrate .