To solve this problem, we need to understand the relationship between ligand field strength and the wavelength of absorption in the visible region for $Co^{3+}$ complexes.$\newline$The wavelength of absorption is inversely proportional to the energy difference $(\Delta)$ between the $t_{2g}$ and $e_g$ orbitals.$\newline$Stronger field ligands increase $\Delta,$ resulting in shorter wavelengths of absorption.$\newline$Let's analyze the ligands in the given complexes:$\newline$• $$H$_2$O$$ is a weak field ligand.$\newline$• $$NH$_3$ is a stronger field ligand than $$H$_2$O$. \newline$• $$en (ethylenediamine)$$ is a bidentate ligand and a stronger field ligand than $$NH$_3. \newline$Therefore, the increasing order of ligand field strength is:$\newline$H$_2$O$<$NH$_3 <$en$\newline$This implies the increasing order of the wavelengths of absorption is:$\newline [$Co(en)$_3]^{3+}, [$Co(NH$_3)_6]^{3+}, [$Co(H$_2$O)$_6]^{3+} \newline$Thus, the correct increasing order for the wavelengths of absorption is:$\newline$Option 4: $[$Co(en)$_3]^{3+}, [$Co(NH$_3)_6]^{3+}, [$Co(H$_2$O)$_6]^{3+}$