The complex that dissolves in water is :

<b>Option A :</b> $\mathrm{Fe}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]_{3}$ (Prussian Blue) <br/><br/>Prussian blue is a very stable and intensely colored compound, but it is not soluble in water. The individual cyanide ligands and the iron ions are bonded together strongly, which results in a crystalline structure that does not easily dissociate into its components in water. <br/><br/><b>Option B :</b> $$\left(\mathrm{NH}_{4}\right)_{3}\left[\mathrm{As}\left(\mathrm{Mo}_{3} \mathrm{O}_{10}\right)_{4}\right]$$ (Ammonium Arseno Molybdate) <br/><br/>In this compound, the arsenic and molybdenum atoms are coordinated in a complex polyatomic anion. Despite the presence of ammonium ions, which are generally soluble, the overall complex has a low solubility due to the strong interactions between arsenic, molybdenum, and oxygen within the anion. <br/><br/><b>Option C :</b> $\mathrm{K}_{3}\left[\mathrm{Co}\left(\mathrm{NO}_{2}\right)_{6}\right]$ <br/><br/>This is a potassium salt of an anionic complex. Although potassium salts are generally soluble in water, the nitrite ligands attached to the cobalt in the anion form a tightly bonded structure, resulting in very poor solubility in water. <br/><br/><b>Option D :</b> $\left[\mathrm{Fe}_{3}(\mathrm{OH})_{2}(\mathrm{OAc})_{6}\right] \mathrm{Cl}$ <br/><br/>This complex is composed of iron atoms coordinated with hydroxide and acetate ligands, and a chloride counterion. Hydroxide and acetate ions are both polar and capable of forming hydrogen bonds with water molecules, which is a key factor in solubility. The chloride ion is also soluble in water. The presence of these ions, combined with the likely overall charge of the complex (since it's paired with a counterion), makes it soluble in water.