What is synergic bonding?

In metal carbonyls, bonding between metal and CO has two components:

• $\sigma$-donation: lone pair on carbon of CO donates to metal.
• $\pi$-back donation: filled metal $d$-orbitals donate electron density into vacant $\pi^*$ orbitals of CO.

When both occur together and reinforce each other, it is called synergic bonding.

Check each given species

We are given:

• $[Cr(CO)_6]$
• $[Mn(CO)_5]$
• $[Mn_2(CO)_{10}]$

For $[Cr(CO)_6]$

This is a metal carbonyl complex. CO is a strong $\pi$-acceptor ligand, so metal-to-ligand $\pi$ back bonding occurs along with $\sigma$ donation.

Hence, $[Cr(CO)_6]$ does exhibit synergic bonding.

For $[Mn(CO)_5]$

This is also a carbonyl complex/radical species with CO ligands. Each CO ligand bonds through the same $\sigma$-donation and $\pi$-back bonding interaction.

Hence, $[Mn(CO)_5]$ does exhibit synergic bonding.

For $[Mn_2(CO)_{10}]$

This is dimanganese decacarbonyl, again a metal carbonyl compound. The Mn-CO bonds involve $\sigma$ donation from CO and $\pi$ back donation from Mn to CO.

Hence, $[Mn_2(CO)_{10}]$ does exhibit synergic bonding.

Total number of complexes showing synergic bonding

All three given species show synergic bonding.

Therefore, the number is: $3$