The complex that dissolves in water is

Let us analyze the solubility of each complex in water:

Option 1: (NH$$_4$$)$$_3$$[AsMo$$_3$$O$$_{10}$$]$$_4$$ - Ammonium arsenomolybdate is an insoluble yellow precipitate (used to test for arsenate ions).

Option 2: Fe$$_4$$[Fe(CN)$$_6$$]$$_3$$ - This is Prussian blue (ferric ferrocyanide), which is an insoluble blue pigment.

Option 3: K$$_3$$[Co(NO$$_2$$)$$_6$$] - Potassium cobaltinitrite is an insoluble yellow precipitate (used as a confirmatory test for K$$^+$$ ions).

Option 4: [Fe$$_3$$(OH)$$_2$$(OAc)$$_6$$]Cl - This is a basic iron(III) acetate complex. It is a cationic complex with Cl$$^-$$ as the counter ion, making it water-soluble.

Therefore, the complex that dissolves in water is [Fe$$_3$$(OH)$$_2$$(OAc)$$_6$$]Cl.