Number of paramagnetic complexes among the following is ________....

To determine the magnetic nature of the given complexes, we analyze the oxidation state, electronic configuration, and the nature of the ligands for each central metal atom/ion:

1. $[\text{MnBr}_4]^{2-}$: Mn$^{2+}$ has a $3d^5$ configuration. Br$^-$ is a weak field ligand. The complex is tetrahedral with $5$ unpaired electrons. (Paramagnetic)

2. $[\text{NiCl}_4]^{2-}$: Ni$^{2+}$ has a $3d^8$ configuration. Cl$^-$ is a weak field ligand. The complex is tetrahedral with $2$ unpaired electrons. (Paramagnetic)

3. $[\text{Ni(CN)}_4]^{2-}$: Ni$^{2+}$ has a $3d^8$ configuration. CN$^-$ is a strong field ligand. The complex is square planar ($dsp^2$) with all electrons paired. (Diamagnetic)

4. $[\text{Ni(CO)}_4]$: Ni$^0$ has a $3d^8 4s^2$ configuration, which becomes $3d^{10}$ in the presence of the strong field ligand CO. The complex is tetrahedral with all electrons paired. (Diamagnetic)

5. $[\text{CoF}_6]^{3-}$: Co$^{3+}$ has a $3d^6$ configuration. F$^-$ is a weak field ligand. The complex is octahedral and high spin ($t_{2g}^4 e_g^2$) with $4$ unpaired electrons. (Paramagnetic)

6. $[\text{Fe(CN)}_6]^{4-}$: Fe$^{2+}$ has a $3d^6$ configuration. CN$^-$ is a strong field ligand. The complex is octahedral and low spin ($t_{2g}^6 e_g^0$) with all electrons paired. (Diamagnetic)

7. $[\text{Mn(CN)}_6]^{3-}$: Mn$^{3+}$ has a $3d^4$ configuration. CN$^-$ is a strong field ligand. The complex is octahedral and low spin ($t_{2g}^4 e_g^0$) with $2$ unpaired electrons. (Paramagnetic)

8. $[\text{Ti(CN)}_6]^{3-}$: Ti$^{3+}$ has a $3d^1$ configuration. The complex is octahedral ($t_{2g}^1 e_g^0$) with $1$ unpaired electron. (Paramagnetic)

9. $[\text{Cu(H}_2\text{O)}_6]^{2+}$: Cu$^{2+}$ has a $3d^9$ configuration. The complex is octahedral ($t_{2g}^6 e_g^3$) with $1$ unpaired electron. (Paramagnetic)

10. $[\text{Co(C}_2\text{O}_4\text{)}_3]^{3-}$: Co$^{3+}$ has a $3d^6$ configuration. Oxalate ($\text{C}_2\text{O}_4^{2-}$) acts as a strong field ligand for Co$^{3+}$. The complex is octahedral and low spin ($t_{2g}^6 e_g^0$) with all electrons paired. (Diamagnetic)

Therefore, the paramagnetic complexes are $[\text{MnBr}_4]^{2-}$, $[\text{NiCl}_4]^{2-}$, $[\text{CoF}_6]^{3-}$, $[\text{Mn(CN)}_6]^{3-}$, $[\text{Ti(CN)}_6]^{3-}$, and $[\text{Cu(H}_2\text{O)}_6]^{2+}$.

The total number of paramagnetic complexes is $6$.

Answer: $6$