$$Mn_2(CO)_{10}$$ is an organometallic compound due to the presence of:

First, we recall the basic definition of an organometallic compound. An organometallic compound is one that contains at least one direct metal-carbon $$\left( \text{M-C} \right)$$ bond, where the carbon is part of an organic group (commonly a ligand such as CO, CH3, C2H5, etc.). This definition is fundamental and is the key criterion chemists use to decide whether any given complex is placed in the organometallic category.

Now let us examine the given molecule $$\mathrm{Mn_2(CO)_{10}}$$ in detail. The formula tells us that there are two manganese atoms and ten carbonyl (CO) ligands.

In a metal carbonyl, each CO ligand is attached to the metal via the carbon atom of the CO group. Symbolically we write a single ligand attachment as

$$\mathrm{Mn} - \mathrm{C} \equiv \mathrm{O}$$

This notation means that the carbon of each carbonyl donates a lone pair to the vacant orbitals of the metal, forming a $$\mathrm{Mn-C}$$ sigma bond, while the $$\mathrm{C \equiv O}$$ unit as a whole also engages in π back-bonding. The critical point for the present question is that there is a direct $$\mathrm{Mn-C}$$ bond in every $$\mathrm{Mn-CO}$$ fragment.

Because

$$\text{Presence of a direct metal-carbon bond} \;\Longrightarrow\; \text{Organometallic nature}$$

and we have just established the existence of ten such $$\mathrm{Mn-C}$$ bonds in $$\mathrm{Mn_2(CO)_{10}},$$ we can confidently classify the molecule as organometallic.

Let us quickly inspect the other bonds mentioned in the options:

• $$\mathrm{Mn-O}$$ bond: In carbonyls, oxygen is not directly bound to the metal, so this bond is absent.
• $$\mathrm{Mn-Mn}$$ bond: $$\mathrm{Mn_2(CO)_{10}}$$ does possess an $$\mathrm{Mn-Mn}$$ bond, but a metal-metal bond alone does not define a compound as organometallic.
• $$\mathrm{C-O}$$ bond: This is purely non-metallic and again is not the defining feature.

The unique feature that confers organometallic status is solely the $$\mathrm{Mn-C}$$ bond.

Hence, the correct answer is Option D.