1. We need the number of bridging carbonyl ligands in $[Mn_2(CO)_{10}]$.

2. First, identify the metal oxidation state:

   • $CO$ is a neutral ligand.
   • The complex is neutral.
   • Therefore, each $Mn$ is in oxidation state $0$.

3. Count electrons around each manganese atom.

   In the usual structure of $[Mn_2(CO)_{10}]$:

   • There is an $Mn-Mn$ single bond.
   • Each $Mn$ is bonded to five terminal $CO$ ligands.

4. Electron count for each $Mn$:

   • $Mn$ is group 7, so $Mn^0$ contributes $7$ electrons.
   • $5$ terminal $CO$ ligands donate $5 \times 2 = 10$ electrons.
   • One $Mn-Mn$ bond contributes $1$ electron to each metal in covalent counting.

   Total for each $Mn$: $7 + 10 + 1 = 18$

5. Thus, each manganese already satisfies the 18-electron rule with five terminal carbonyls and one metal-metal bond. There is no need for any bridging $CO$ ligand.

6. Therefore, in $[Mn_2(CO)_{10}]$, all 10 carbonyl ligands are terminal.

Hence, the number of bridging $CO$ ligands is: $\boxed{0}$