Step 1: Write the formula and identify the components.
The complex is $[Co(en)_3]_2(SO_4)_3$. Here $en$ is ethylenediamine (a neutral ligand), and $SO_4^{2-}$ is the counter ion outside the coordination sphere.
Step 2: Calculate the total charge from sulphate counter ions.
There are 3 sulphate ions, each with charge $-2$: \[ \text{Total negative charge} = 3 \times (-2) = -6 \]
Step 3: Find the charge each complex cation must carry.
For electrical neutrality, the two complex cations must together carry charge $+6$. \[ 2 \times [Co(en)_3]^{n+} = +6 \Rightarrow \text{each cation has charge } +3 \]
Step 4: Assign the oxidation state of Co.
Ethylenediamine ($en$) is a neutral ligand (charge = 0). So all $+3$ charge comes from Co. \[ x + 3(0) = +3 \Rightarrow x = +3 \]
Step 5: Cross-check electrical neutrality.
$Co(+3)$ with three neutral $en$ ligands gives cation charge $+3$. Two such cations give $+6$, balanced by three $SO_4^{2-}$ giving $-6$. The compound is neutral. Verified!
Step 6: State the final answer.
The oxidation number of cobalt in $[Co(en)_3]_2(SO_4)_3$ is $+3$. \[ \boxed{+3} \]