1. Find oxidation state and electronic configuration of the metal ion in each complex

• In $[\mathrm{FeF}_6]^{3-}$:

  Let oxidation state of Fe be $x$. $x+6(-1)=-3 \Rightarrow x=+3$ So, $\mathrm{Fe}^{3+}: [\mathrm{Ar}]3d^5$

• In $[\mathrm{CoF}_6]^{3-}$: $x+6(-1)=-3 \Rightarrow x=+3$ So, $\mathrm{Co}^{3+}: [\mathrm{Ar}]3d^6$

• In $[\mathrm{Ni}(\mathrm{CO})_4]$: CO is neutral, so Ni is in oxidation state $0$. $\mathrm{Ni}: [\mathrm{Ar}]3d^8 4s^2$ In complexes, it is treated as $d^{10}$ for $\mathrm{Ni}^0$.

• In $[\mathrm{Ni}(\mathrm{CN})_4]^{2-}$: $x+4(-1)=-2 \Rightarrow x=+2$ So, $\mathrm{Ni}^{2+}: [\mathrm{Ar}]3d^8$

2. Determine ligand strength and spin state

• $\mathrm{F}^-$ is a weak field ligand $\Rightarrow$ high-spin complexes.
• CO and CN$^-$ are strong field ligands $\Rightarrow$ pairing occurs.

3. Count unpaired electrons in each complex

(i) $[\mathrm{FeF}_6]^{3-}$

• Metal ion: $\mathrm{Fe}^{3+} = d^5$
• Octahedral with weak field ligand $\mathrm{F}^-$
• High-spin configuration: $t_{2g}^3 e_g^2$
• Number of unpaired electrons $=5$

(ii) $[\mathrm{CoF}_6]^{3-}$

• Metal ion: $\mathrm{Co}^{3+} = d^6$
• Octahedral with weak field ligand $\mathrm{F}^-$
• High-spin configuration: $t_{2g}^4 e_g^2$
• Number of unpaired electrons $=4$

(iii) $[\mathrm{Ni}(\mathrm{CO})_4]$

• $\mathrm{Ni}^0 = d^{10}$
• Tetrahedral complex
• $d^{10}$ means all orbitals are filled
• Number of unpaired electrons $=0$

(iv) $[\mathrm{Ni}(\mathrm{CN})_4]^{2-}$

• $\mathrm{Ni}^{2+}=d^8$
• CN$^-$ is strong field ligand
• This complex is square planar with paired electrons
• Number of unpaired electrons $=0$

4. Arrange in decreasing order of unpaired electrons

$[\mathrm{FeF}_6]^{3-} \; (5) \, > \, [\mathrm{CoF}_6]^{3-} \; (4) \, > \, [\mathrm{Ni}(\mathrm{CN})_4]^{2-} \; (0) \, = \, [\mathrm{Ni}(\mathrm{CO})_4] \; (0)$

This matches:

$\boxed{[\mathrm{FeF}_6]^{3-} > [\mathrm{CoF}_6]^{3-} > [\mathrm{Ni}(\mathrm{CN})_4]^{2-} = [\mathrm{Ni}(\mathrm{CO})_4]}$

So, the correct option is C.

5. Comparison with stored answer

Stored correct answer: C

Our derived answer: C

They agree.