Number of paramagnetic complexes among the following is __________.
$$[\text{MnBr}_4]^{2-}$$, $$[\text{NiCl}_4]^{2-}$$, $$[\text{Ni(CN)}_4]^{2-}$$, $$[\text{Ni(CO)}_4]$$, $$[\text{CoF}_6]^{3-}$$, $$[\text{Fe(CN)}_6]^{4-}$$, $$[\text{Mn(CN)}_6]^{3-}$$, $$[\text{Ti(CN)}_6]^{3-}$$, $$[\text{Cu(H}_2\text{O)}_6]^{2+}$$, $$[\text{Co(C}_2\text{O}_4)_3]^{3-}$$

For determining the number of paramagnetic complexes, we examine the electronic configuration of the central metal atom/ion and the nature of the ligands (Strong Field vs. Weak Field) using Crystal Field Theory (CFT). Complexes with unpaired electrons are paramagnetic.

1. $$[MnBr_4]^{2-}$$: $$Mn^{2+}$$ $$(3d^{5}$$). $$Br^{-}$$ is a weak field ligand forming a tetrahedral complex. The configuration is $$e^{2}t_{2}^{3}$$ $$\rightarrow $$ 5 unpaired electrons (Paramagnetic).
   
   
2. $$[NiCl_4]^{2-}$$: $$Ni^{2+}$$ $$(3d^{8})$$. $$Cl^{-}$$ is a weak field ligand forming a tetrahedral complex. The configuration is $$e^{4}t_{2}^{4}$$ $$\rightarrow $$ 2 unpaired electrons (Paramagnetic).
   
   
3. $$[Ni(CN)_4]^{2-}$$: $$Ni^{2+}$$ $$(3d^{8})$$. $$CN^{-}$$ is a strong field ligand causing pairing in a square-planar geometry. The configuration leaves 0 unpaired electrons (Diamagnetic).
   
   
4. $$[Ni(CO)_4]$$: $$Ni^{0}$$ $$(3d^{8}4s^{2})$$. $$CO$$ is a very strong field ligand that forces $$4s$$ electrons to pair up into the $$3d$$ subshell, making it a fully filled $$3d^{10}$$ system $$\rightarrow $$ 0 unpaired electrons (Diamagnetic).
   
   
5. $$[CoF_6]^{3-}$$: $$Co^{3+}$$ $$(3d^{6})$$. $$F^{-}$$ is a weak field ligand forming a high-spin octahedral complex. The configuration is $$t_{2g}^{4}e_{g}^{2}$$ $$\rightarrow$$ 4 unpaired electrons (Paramagnetic).
   
   
6. $$[Fe(CN)_6]^{4-}$$: $$Fe^{2+}$$ $$(3d^{6})$$. $$CN^{-}$$ is a strong field ligand forming a low-spin octahedral complex. The configuration is $$t_{2g}^{6}e_{g}^{0}$$ $$\rightarrow $$ 0 unpaired electrons (Diamagnetic).
   
   
7. $$[Mn(CN)_6]^{3-}$$: $$Mn^{3+}$$ $$(3d^{4})$$. $$CN^{-}$$ is a strong field ligand forming a low-spin octahedral complex. The configuration is $$t_{2g}^{4}e_{g}^{0}$$ $$\rightarrow $$ 2 unpaired electrons (Paramagnetic).
   
   
8. $$[Ti(CN)_6]^{3-}$$: $$Ti^{3+}$$ $$(3d^{1})$$. Any $$d^{1}$$ system in an octahedral environment will have a $$t_{2g}^{1}e_{g}^{0}$$ configuration $$\rightarrow $$ 1 unpaired electron (Paramagnetic).
   
   
9. $$[Cu(H_2O)_6]^{2+}$$: $$Cu^{2+}$$ $$(3d^{9})$$. In an octahedral crystal field, a $$d^{9}$$ system has the configuration $$t_{2g}^{6}e_{g}^{3}$$ $$\rightarrow $$ 1 unpaired electron (Paramagnetic).
   
   
10. $$[Co(C_2O_4)_3]^{3-}$$: $$Co^{3+}$$ $$(3d^{6})$$. Oxalate $$(C_{2}O_{4}^{2-})$$ acts as a strong field chelating ligand with $$Co^{3+}$$, forcing all electrons to pair up to form a low-spin complex $$(t_{2g}^{6}e_{g}^{0})$$ $$\rightarrow $$ 0 unpaired electrons (Diamagnetic).
    
    

The number of paramagnetic complexes among the given list is 6.