To determine the IUPAC name of the complex $[Ag(H_2O)_2][Ag(CN)_2],$ we need to follow the IUPAC nomenclature rules for coordination compounds.$\newline$1. Identify the cation and anion in the complex:$\newline$• The complex is composed of two parts: $[Ag(H_2O)_2]^+$ and $[Ag(CN)_2]^-. \newline$• $[Ag(H_2O)_2]^+$ is the cation, and $[Ag(CN)_2]^-$ is the anion.$\newline$2. Naming the cation:$\newline$• The cation $[Ag(H_2O)_2]^+$ contains silver coordinated with two water molecules.$\newline$• In naming, the ligands are named first followed by the metal.$\newline$• Water as a ligand is called 'aqua'.$\newline$• Since there are two water molecules, we use the prefix 'di'.$\newline$• The oxidation state of silver in $[Ag(H_2O)_2]^+$ is +1.$\newline$• Therefore, the cation is named 'diaquasilver(I)'.$\newline$3. Naming the anion:$\newline$• The anion $[Ag(CN)_2]^-$ contains silver coordinated with two cyanide ions.$\newline$• Cyanide as a ligand is called 'cyanido'.$\newline$• Since there are two cyanide ions, we use the prefix 'di'.$\newline$• The oxidation state of silver in $[Ag(CN)_2]^-$ is +1.$\newline$• Therefore, the anion is named 'dicyanidoargentate(I)'.$\newline$4. Combining the names:$\newline$• The cation name is written first followed by the anion name.$\newline$• Thus, the IUPAC name of the complex is 'Diaquasilver(I) dicyanidoargentate(I)'.$\newline$This corresponds to Option 4.