Step 1: Recall definition of coordination number.
The coordination number of a central metal atom in a complex is the number of ligand donor atoms directly bonded to it.
Step 2: Identify ligands in [Cu(CN)$_4$]$^{3-$.}
- The ligand present is CN$^-$.
- Each CN$^-$ ligand donates one pair of electrons through its carbon atom.
Step 3: Count the number of ligands.
There are 4 cyanide ligands attached to Cu. Therefore, the number of Cu–C bonds formed = 4.
Step 4: Conclusion.
Hence, the coordination number of Cu in [Cu(CN)$_4$]$^{3-}$ is 4.
\[
\boxed{4}
\]
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