The d-orbital electronic configuration of the complex among $$[Co(en)_3]^{3+}$$, $$[CoF_6]^{3-}$$, $$[Mn(H_2O)_6]^{2+}$$ and $$[Zn(H_2O)_6]^{2+}$$ that has the highest CFSE is :

First, write the outer electronic configuration of each metal ion (remember that electrons are removed first from the 4s orbital, then from 3d).
Co (atomic no. 27): [Ar] 3d$$^{7}$$ 4s$$^{2}$$ → Co$$^{3+}$$ : [Ar] 3d$$^{6}$$
Mn (atomic no. 25): [Ar] 3d$$^{5}$$ 4s$$^{2}$$ → Mn$$^{2+}$$ : [Ar] 3d$$^{5}$$
Zn (atomic no. 30): [Ar] 3d$$^{10}$$ 4s$$^{2}$$ → Zn$$^{2+}$$ : [Ar] 3d$$^{10}$$

Next, decide whether the complex is high-spin or low-spin. This depends on the ligand field strength (spectrochemical series).
en (ethylenediamine) is a strong-field ligand, F$$^-$$ and H$$_2$$O are weak-field ligands.

Case 1: $$[Co(en)_3]^{3+}$$
Ligand: en → strong field
Ion: Co$$^{3+}$$ → d$$^{6}$$
Strong field ⇒ low-spin (pairing preferred).
d$$^{6}$$ low-spin octahedral configuration: $$t_{2g}^{6}e_g^{0}$$

Case 2: $$[CoF_6]^{3-}$$
Ligand: F$$^-$$ → weak field
Ion: Co$$^{3+}$$ → d$$^{6}$$
Weak field ⇒ high-spin (no extra pairing).
d$$^{6}$$ high-spin octahedral configuration: $$t_{2g}^{4}e_g^{2}$$

Case 3: $$[Mn(H_2O)_6]^{2+}$$
Ligand: H$$_2$$O → weak field
Ion: Mn$$^{2+}$$ → d$$^{5}$$
Weak field ⇒ high-spin.
d$$^{5}$$ high-spin octahedral configuration: $$t_{2g}^{3}e_g^{2}$$

Case 4: $$[Zn(H_2O)_6]^{2+}$$
Ligand: H$$_2$$O → weak field
Ion: Zn$$^{2+}$$ → d$$^{10}$$
All d-orbitals already filled irrespective of field strength.
Configuration: $$t_{2g}^{6}e_g^{4}$$

Now calculate the crystal field stabilization energy (CFSE) for each octahedral configuration.
Formula: CFSE = number in $$t_{2g}$$ × $$(-0.4\Delta_o)$$ + number in $$e_g$$ × $$(+0.6\Delta_o)$$

$$t_{2g}^{6}e_g^{0}$$ : CFSE = $$6(-0.4\Delta_o)= -2.4\Delta_o$$
$$t_{2g}^{4}e_g^{2}$$ : CFSE = $$4(-0.4\Delta_o)+2(+0.6\Delta_o)= -0.4\Delta_o$$
$$t_{2g}^{3}e_g^{2}$$ : CFSE = $$3(-0.4\Delta_o)+2(+0.6\Delta_o)= 0$$
$$t_{2g}^{6}e_g^{4}$$ : CFSE = $$6(-0.4\Delta_o)+4(+0.6\Delta_o)= 0$$

The most negative value (largest magnitude of stabilisation) is $$-2.4\Delta_o$$ for $$t_{2g}^{6}e_g^{0}$$.

Therefore the complex with the highest CFSE is $$[Co(en)_3]^{3+}$$, and its d-orbital electronic configuration is $$t_{2g}^{6}e_g^{0}$$.

Correct option: Option A