**Core Concept**
The cough reflex on scratching the floor of the external auditory meatus is a clinical manifestation of the auriculotemporal nerve's connection to the vagus nerve. This phenomenon is known as the Arnold's reflex, which is a polysynaptic reflex arc that involves the activation of the vagus nerve.

**Why the Correct Answer is Right**
The auriculotemporal nerve, a branch of the mandibular division of the trigeminal nerve, carries fibers that synapse with the vagus nerve in the medulla oblongata. The vagus nerve, in turn, stimulates the cough center in the medulla, leading to the characteristic cough reflex. This reflex is thought to be an evolutionary remnant of the gag reflex, which helps protect the airway during swallowing.

**Why Each Wrong Option is Incorrect**
**Option A:** The glossopharyngeal nerve is involved in swallowing and gag reflexes but is not directly responsible for the cough reflex on scratching the external auditory meatus.
**Option B:** The facial nerve primarily controls facial expressions and taste, and is not involved in this reflex arc.
**Option C:** The trigeminal nerve, while involved in the auriculotemporal nerve, is not the primary nerve responsible for the cough reflex.

**Clinical Pearl / High-Yield Fact**
Arnold's reflex can be used as a clinical indicator of upper motor neuron lesions, particularly those affecting the brainstem or spinal cord.

**Correct Answer: C. Auriculotemporal nerve.**