1. Understand what is asked

We need the total number of bridging carbonyl ligands in the two complexes:

• $\mathrm{W(CO)_6}$
• $\mathrm{Mn_2(CO)_{10}}$

A bridging carbonyl is a $\mu$-CO ligand that connects two metal atoms.

2. Analyse $\mathrm{W(CO)_6}$

$\mathrm{W(CO)_6}$ is a mononuclear metal carbonyl complex.

• It contains only one tungsten atom.
• Since there is only one metal center, a carbonyl ligand cannot bridge between two metals.

Therefore, number of bridging carbonyls in $\mathrm{W(CO)_6}$ is:

$0$

3. Analyse $\mathrm{Mn_2(CO)_{10}}$

$\mathrm{Mn_2(CO)_{10}}$ is a binuclear metal carbonyl.

Its structure consists of:

• an $\mathrm{Mn-Mn}$ bond
• 10 terminal CO ligands
• no bridging CO ligands

This is the standard structure of dimanganese decacarbonyl.

Therefore, number of bridging carbonyls in $\mathrm{Mn_2(CO)_{10}}$ is:

$0$

4. Find the sum

$0 + 0 = 0$

5. Final answer

The sum of bridging carbonyls is:

$\boxed{0}$

6. Comparison with stored correct answer

Stored correct answer = $0$

Our derived answer = $0$

So, the answer matches the stored correct answer.