To find the sum of the oxidation states of the metals in $\mathrm{Fe(CO)}_{5}$, $\mathrm{VO}^{2+}$, and $\mathrm{WO}_{3}$, we first need to find the oxidation states of the metals in each compound.

1. $\mathrm{Fe(CO)}_{5}$: The compound $\mathrm{CO}$ is a neutral ligand, so it does not contribute to the oxidation state of the metal. Hence, the oxidation state of Fe in $\mathrm{Fe(CO)}_{5}$ is 0.




2. $\mathrm{VO}^{2+}$: The compound $\mathrm{O}$ has an oxidation state of -2, and since the overall charge of the compound is +2, the oxidation state of V must be +4 to balance this out.




3. $\mathrm{WO}_{3}$: The compound $\mathrm{O}$ has an oxidation state of -2, and since there are three oxygen atoms, the total oxidation state contributed by oxygen is -6. To balance this out, the oxidation state of W must be +6.

Therefore, the sum of the oxidation states of the metals in these three compounds is $0 + 4 + 6 = 10$.