Let us deduce the formula from the given IUPAC name 'chlorodiaquatriamminecobalt(III) chloride':

1. Central metal: Cobalt (Co) in +3 oxidation state.
2. Ligands in the coordination sphere (alphabetical order in name):

• 'chloro' indicates one $Cl^-$ ligand.
• 'diaqua' indicates two $H_2O$ ligands.
• 'triammine' indicates three $NH_3$ ligands. So, the coordination entity is $[CoCl(NH_3)_3(H_2O)_2]$.

3. Charge on the coordination sphere: $x = (+3) + (-1) + 2(0) + 3(0) = +2$.
4. Counter ion: 'chloride' indicates $Cl^-$ outside the coordination sphere.
5. To balance the +2 charge of the complex cation, two chloride ions are required outside the bracket. Therefore, the complete formula is $[CoCl(NH_3)_3(H_2O)_2]Cl_2$.