**Core Concept**
The metacarpophalangeal (MCP) joint is a complex hinge joint in the hand, allowing flexion and extension movements. The nerve responsible for controlling the muscles involved in MCP extension is a branch of the radial nerve, which primarily controls wrist and finger extension.

**Why the Correct Answer is Right**
The radial nerve supplies the extensor digitorum muscle, which is responsible for extending the MCP joint. When the radial nerve is injured, the extensor digitorum muscle is paralyzed, leading to an inability to extend the MCP joint. This is due to the absence of the radial nerve's motor innervation to the extensor digitorum muscle.

**Why Each Wrong Option is Incorrect**
* **Option A:** The median nerve primarily controls flexion of the MCP joint, not extension.
* **Option B:** The ulnar nerve controls adduction and abduction of the fingers, but not extension of the MCP joint.
* **Option C:** The musculocutaneous nerve primarily controls elbow flexion and forearm supination, not MCP extension.

**Clinical Pearl / High-Yield Fact**
The radial nerve is often injured in cases of radial fracture (Colles' fracture) or radial nerve compression at the elbow or wrist, leading to a characteristic "wrist drop" or inability to extend the wrist and fingers.

**Correct Answer: C. Ulnar nerve.**