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The metal carbon bond in metal carbonyls possesses both $\sigma$ and $\pi$ character.
Reason: The ligand to metal bond is a $\pi$ bond and metal to ligand bond is a $\sigma$ bond.
Detailed Solution
The M-C bond in metal carbonyls has both $\sigma$ and $\pi$ character, so the assertion is correct. The ligand-to-metal bond is a $\sigma$ bond (donation of lone pair of CO to metal) and the metal-to-ligand bond is a $\pi$ bond (back donation into $\pi^*$ of CO). The reason states the reverse, so it is incorrect.