Number of complexes which will exhibit synergic bonding amongst, $$[Cr(CO)_6], [Mn(CO)_5]$$ and $$[Mn_2(CO)_{10}]$$ is ______

We need to determine how many of the given complexes exhibit synergic bonding.

Synergic bonding occurs when a ligand simultaneously acts as a sigma ($$\sigma$$) donor and a pi ($$\pi$$) acceptor, donating a lone pair to the metal and receiving electron density back into its empty antibonding orbitals, a process known as pi back‐bonding. The CO ligand is a classic example: it forms a sigma bond by donating its lone pair to the metal and accepts electron density from filled metal d‐orbitals into its empty $$\pi^*$$ antibonding orbitals.

In the complex $$[Cr(CO)_6]$$, CO ligands bond to Cr and participate in both $$\sigma$$-donation and $$\pi$$-back bonding, so synergic bonding is present. Similarly, $$[Mn(CO)_5]$$ contains CO ligands bonded to Mn with the same dual interaction, and $$[Mn_2(CO)_{10}]$$ features CO ligands on each Mn atom (along with a Mn-Mn bond), where each Mn-CO interaction also involves synergic bonding.

Since all three complexes have CO ligands that exhibit simultaneous $$\sigma$$ donation and $$\pi$$ back‐bonding with the metal centers, the total number of complexes exhibiting synergic bonding is 3.