Step 1: Recall branches of the facial nerve (CN VII).
The facial nerve gives off several important branches within the facial canal and after exiting the stylomastoid foramenThese include posterior auricular nerve, nerve to stapedius, and chorda tympani
Step 2: Analyze each option.
(A) Posterior auricular: A branch of the facial nerveSupplies occipital belly of occipitofrontalis and auricular muscles
(B) Nerve to stapedius: Branch of facial nerve within the facial canalSupplies stapedius muscle
(D) Chorda tympani: Branch of facial nerveCarries taste from anterior two-thirds of tongue and parasympathetic fibers
Step 3: Identify the exception.
(C) Deep petrosal nerve: This is not a branch of the facial nerveIt carries sympathetic fibers from the internal carotid plexus and joins the greater petrosal nerve to form the nerve of the pterygoid canal
Step 4: Conclusion.
Therefore, the deep petrosal nerve is not supplied by the facial nerve