1. We need the oxidation state of the metal in each species:

   • $\mathrm{Fe(CO)_5}$
   • $\mathrm{VO^{2+}}$
   • $\mathrm{WO_3}$

2. For $\mathrm{Fe(CO)_5}$:

   • Carbon monoxide, $\mathrm{CO}$, is a neutral ligand.
   • Total charge on the complex is $0$.
   • Let oxidation state of Fe be $x$. $x + 5(0) = 0$ $x = 0$ So, oxidation state of Fe is $0$.

3. For $\mathrm{VO^{2+}}$:

   • Oxygen usually has oxidation state $-2$.
   • Overall charge is $+2$.
   • Let oxidation state of V be $x$. $x + (-2) = +2$ $x = +4$ So, oxidation state of V is $+4$.

4. For $\mathrm{WO_3}$:

   • Oxygen has oxidation state $-2$.
   • Let oxidation state of W be $x$. $x + 3(-2) = 0$ $x - 6 = 0$ $x = +6$ So, oxidation state of W is $+6$.

5. Sum of oxidation states of the metals: $0 + 4 + 6 = 10$

Therefore, the required integer answer is $10$.