Step 1: Understanding the Concept:
Hybridization in coordination compounds is determined by the coordination number and the nature of the ligands (Strong field vs Weak field), which influences electron pairing in the $d$-orbitals.
Step 3: Detailed Explanation:
1. \([Ni(CO)_{4]\)}: Ni is in 0 oxidation state ($3d^{8}4s^{2}$). $CO$ is a very strong ligand and causes pairing of $4s$ electrons into $3d$, resulting in $3d^{10}$. The empty $4s$ and $4p$ orbitals hybridize. Hybridization: \(sp^{3}\) (A-2).
2. \([Ni(CN)_{4]^{2-}\)}: Ni is in $+2$ state ($3d^{8}$). $CN^{-}$ is a strong field ligand and causes pairing of the two unpaired $d$ electrons. This leaves one $3d$ orbital vacant. Hybridization: \(dsp^{2}\) (B-1).
3. \([Ni(NH_{3})_{6]^{2+}\)}: Ni is in $+2$ state ($3d^{8}$). Coordination number is 6. With 8 electrons, it cannot form an inner orbital complex ($d^{2}sp^{3}$) as only two $d$-orbitals are needed. It uses outer $d$-orbitals. Hybridization: \(sp^{3}d^{2}\) (C-4).
4. \([Fe(CN)_{6]^{3-}\)}: Fe is in $+3$ state ($3d^{5}$). $CN^{-}$ is a strong field ligand and causes pairing, leaving two $3d$ orbitals vacant for hybridization. Hybridization: \(d^{2}sp^{3}\) (D-3).
Step 4: Final Answer:
The correct match is A-2, B-1, C-4, D-3.