The spin only magnetic moments (in BM) for free Ti$$^{3+}$$, V$$^{2+}$$ and Sc$$^{3+}$$ ions respectively are (At.No. Sc : 21, Ti : 22, V : 23)

For transition-metal ions the spin-only magnetic moment $$\mu_{\text{spin}}$$ (in Bohr Magnetons, BM) is obtained from the well-known formula

$$\mu_{\text{spin}}=\sqrt{n\,(n+2)}\;\text{BM},$$

where $$n$$ is the number of unpaired electrons present in the ion. We must therefore write the electronic configuration of each ion, count the unpaired electrons, and then substitute the value of $$n$$ in the above expression.

Ion 1 : Ti$$^{3+}$$

Neutral titanium (Ti, $$Z=22$$) has the configuration

$$\text{Ti} : [\text{Ar}]\,3d^{2}\,4s^{2}.$$

For the $$+3$$ oxidation state we remove two electrons first from the $$4s$$ orbital and one electron from the $$3d$$ orbital:

$$\text{Ti}^{3+} : [\text{Ar}]\,3d^{1}.$$

This single $$3d$$ electron is unpaired, so $$n=1$$. Substituting in the formula,

$$\mu_{\text{spin}}=\sqrt{1\,(1+2)}=\sqrt{3}=1.732\;\text{BM}\;(\text{≈}1.73\;\text{BM}).$$

Ion 2 : V$$^{2+}$$

Neutral vanadium (V, $$Z=23$$) has the configuration

$$\text{V} : [\text{Ar}]\,3d^{3}\,4s^{2}.$$

In the $$+2$$ state we remove the two $$4s$$ electrons:

$$\text{V}^{2+} : [\text{Ar}]\,3d^{3}.$$

The three $$3d$$ electrons occupy different orbitals with parallel spins (Hund’s rule), so all three are unpaired. Thus $$n=3$$. Putting this into the formula,

$$\mu_{\text{spin}}=\sqrt{3\,(3+2)}=\sqrt{15}=3.873\;\text{BM}\;(\text{≈}3.87\;\text{BM}).$$

Ion 3 : Sc$$^{3+}$$

Neutral scandium (Sc, $$Z=21$$) has the configuration

$$\text{Sc} : [\text{Ar}]\,3d^{1}\,4s^{2}.$$

Removing three electrons (two from $$4s$$ and one from $$3d$$) for the $$+3$$ state gives

$$\text{Sc}^{3+} : [\text{Ar}],$$

which is a noble-gas core with no electrons in the valence shell. Therefore $$n=0$$ and

$$\mu_{\text{spin}}=\sqrt{0\,(0+2)}=0\;\text{BM}.$$

Collecting the results in the same order as asked (Ti$$^{3+}$$, V$$^{2+}$$, Sc$$^{3+}$$):

$$1.73\;\text{BM},\;3.87\;\text{BM},\;0\;\text{BM}.$$

These values match exactly with the entries of Option B.

Hence, the correct answer is Option B.