1. Identify oxidation state and d-electron count

All complexes are octahedral with strong field ligand $\text{CN}^-$. We compare their octahedral crystal field splitting $\Delta_o$.

• I. $[\text{Mn}(\text{CN})_6]^{3-}$

  Let oxidation state of Mn be $x$: $x+6(-1)=-3 \Rightarrow x=+3$ So Mn is $\text{Mn}^{3+}$, i.e. $3d^4$.

• II. $[\text{Co}(\text{CN})_6]^{4-}$ $x+6(-1)=-4 \Rightarrow x=+2$ So Co is $\text{Co}^{2+}$, i.e. $3d^7$.

• III. $[\text{Fe}(\text{CN})_6]^{4-}$ $x+6(-1)=-4 \Rightarrow x=+2$ So Fe is $\text{Fe}^{2+}$, i.e. $3d^6$.

• IV. $[\text{Fe}(\text{CN})_6]^{3-}$ $x+6(-1)=-3 \Rightarrow x=+3$ So Fe is $\text{Fe}^{3+}$, i.e. $3d^5$.

2. Use the trend of $\Delta_o$

For a given ligand, $\Delta_o$ generally:

• increases with increasing oxidation state of the metal,
• increases across the period from left to right for 3d metals.

Since all ligands are the same ($\text{CN}^-$), compare mainly using metal ion identity and oxidation state.

3. Pairwise comparison

• Between III and IV: $[\text{Fe}(\text{CN})_6]^{3-} \text{ has Fe}^{3+} > \text{Fe}^{2+}$ so $\Delta_o(\text{IV}) > \Delta_o(\text{III})$

• Between I and IV: Both are $+3$ oxidation state, but Fe is to the right of Mn in the 3d series, so $\Delta_o(\text{IV}) > \Delta_o(\text{I})$

• Between III and II: Both are $+2$ oxidation state, but Co is to the right of Fe, so $\Delta_o(\text{II}) > \Delta_o(\text{III})$

4. Construct overall increasing order

Thus the order becomes $\text{I} < \text{III} < \text{II} < \text{IV}$

5. Match with the given options

The derived order is: $[\text{Mn}(\text{CN})_6]^{3-} < [\text{Fe}(\text{CN})_6]^{4-} < [\text{Co}(\text{CN})_6]^{4-} < [\text{Fe}(\text{CN})_6]^{3-}$

This order is not present exactly in the options.

Now check the stored answer A: $\text{I} < \text{II} < \text{IV} < \text{III}$ This would imply $\Delta_o(\text{IV}) < \Delta_o(\text{III})$, i.e. Fe$^{3+}$ has smaller splitting than Fe$^{2+}$ for the same ligand, which is incorrect.

Hence the stored answer does not agree with crystal field trends.

6. Conclusion

The correct increasing order based on $\Delta_o$ should be: $\boxed{\text{I} < \text{III} < \text{II} < \text{IV}}$

Since no option matches exactly, the stored answer appears incorrect.