**Core Concept**
The question is testing the student's knowledge of the unique characteristics of the cranial nerves that innervate the muscles of the face and head. Specifically, it is asking about the cranial nerve that has a contralateral distribution, meaning it supplies a muscle on the opposite side of the body.

**Why the Correct Answer is Right**
The correct answer is the **Oculomotor nerve (III)**, which is responsible for innervating the medial rectus muscle of the eye. The medial rectus muscle is responsible for adduction of the eye, bringing it towards the midline of the body. Since the eyes are on opposite sides of the head, the medial rectus muscle on one side must be innervated by the oculomotor nerve on the opposite side of the head, allowing for coordinated eye movement.

**Why Each Wrong Option is Incorrect**

* **Option A:** The Trochlear nerve (IV) primarily innervates the superior oblique muscle, which is responsible for intorsion and depression of the eye. It does not have a contralateral distribution.
* **Option B:** The Abducens nerve (VI) primarily innervates the lateral rectus muscle, which is responsible for abduction of the eye. It also does not have a contralateral distribution.
* **Option D:** The Trigeminal nerve (V) is a sensory nerve that provides sensation to the face, and does not have a motor component that supplies a contralateral muscle.

**Clinical Pearl / High-Yield Fact**
Remember that the oculomotor nerve (III) is the only cranial nerve that has a contralateral distribution, allowing for coordinated eye movement. This can be a useful fact to remember when evaluating patients with eye movement disorders.

**Correct Answer:** C. Oculomotor nerve (III)