## **Core Concept**
The **third branchial arch**, also known as the third pharyngeal arch, is one of the embryological structures that develop into various adult head and neck features. Each branchial arch is associated with specific cranial nerves, muscles, and skeletal elements. The nerve associated with the third branchial arch is a key aspect of head and neck anatomy.

## **Why the Correct Answer is Right**
The **glossopharyngeal nerve (CN IX)** is the nerve of the third branchial arch. This nerve is responsible for swallowing and taste sensation from the posterior one-third of the tongue. It also provides motor innervation to the stylopharyngeus muscle, which is derived from the third branchial arch.

## **Why Each Wrong Option is Incorrect**
* **Option A:** The **trigeminal nerve (CN V)** is associated with the first branchial arch, not the third. It provides sensory innervation to the face and motor innervation to the muscles of mastication.
* **Option B:** The **vagus nerve (CN X)** is associated with the fourth and sixth branchial arches. It has a wide range of functions, including parasympathetic innervation of various organs, motor control of the larynx and pharynx, and sensory innervation of the larynx and viscera.
* **Option D:** The **facial nerve (CN VII)** is associated with the second branchial arch. It controls the muscles of facial expression and provides taste sensation from the anterior two-thirds of the tongue.

## **Clinical Pearl / High-Yield Fact**
A useful clinical correlation to remember is that the **glossopharyngeal nerve (CN IX)**, being the nerve of the third branchial arch, is involved in the **gag reflex**. When the posterior tongue or pharynx is stimulated, CN IX carries the afferent signal to the brainstem, which then sends an efferent signal through the vagus nerve (CN X) to the pharyngeal muscles, inducing a gag response.

## **Correct Answer:** C. Glossopharyngeal nerve.