**Core Concept**
The posterior belly of the digastric muscle is a part of the extrinsic muscles of the tongue, which play a crucial role in the movement of the tongue. The posterior belly of the digastric muscle arises from the digastric fossa on the inner surface of the mandible and is innervated by a cranial nerve.

**Why the Correct Answer is Right**
The posterior belly of the digastric muscle is supplied by the mylohyoid nerve, a branch of the mandibular division of the trigeminal nerve (cranial nerve V3). The mylohyoid nerve arises from the inferior alveolar nerve and provides motor innervation to the posterior belly of the digastric muscle. This innervation allows for the coordinated movement of the tongue and the jaw.

**Why Each Wrong Option is Incorrect**
**Option A:** The facial nerve, primarily responsible for motor functions of facial expression, does not innervate the posterior belly of the digastric muscle.
**Option B:** The lingual nerve, a branch of the mandibular division of the trigeminal nerve, provides sensory innervation to the tongue but does not innervate the posterior belly of the digastric muscle.
**Option C:** The hypoglossal nerve, responsible for motor innervation of the tongue, does not innervate the posterior belly of the digastric muscle.

**Clinical Pearl / High-Yield Fact**
The digastric muscle is unique in that it has two bellies (anterior and posterior) with different innervations, making it an important landmark for cranial nerve identification during clinical examination.

**Correct Answer: C. The hypoglossal nerve does not supply the posterior belly of the digastric muscle. It is supplied by the mylohyoid nerve, a branch of the mandibular division of the trigeminal nerve (cranial nerve V3).**