Why does the central atom in \[\left[ {Fe{{\left( {Co} \right)}_5}} \right]\] has a zero oxidation state?

Hint- In order to deal with this question, we will proceed by assuming the oxidation number of Fe as a variable and we will use the basic concept of chemistry that the sum of total oxidation number is zero.

Complete answer:
Given compound is \[\left[ {Fe{{\left( {Co} \right)}_5}} \right]\]
Let’s assume the oxidation number of Fe be $x$ .
We know that in a compound the sum of total oxidation numbers is zero. If there is any ion then we consider that.
Here CO is a neutral compound having an oxidation number of zero. C shows an oxidation number of +2 whereas oxygen shows -2. That is why CO has zero oxidation number.
So by definition, (as the sum of total oxidation number is zero and also there is no external charge)
\[
   \Rightarrow x + \left( {2 + \left( { - 2} \right)} \right) \times 5 = 0 \\
   \Rightarrow x + 0 \times 5 = 0 \\
   \Rightarrow x + 0 = 0 \\
   \Rightarrow x = 0 \\
 \]
We get $x = 0$ .

Hence, the oxidation number of Fe in this compound is zero.

Note- Oxidation state is a number assigned to an element in a chemical combination which represents the number of electrons lost (or gained, if the number is negative), by an atom of that element in the compound. The oxidation state, sometimes referred to as oxidation number, describes the degree of oxidation of an atom in a chemical compound.