**Core Concept:** The radial nerve is a mixed nerve that supplies motor and sensory functions to the forearm and hand. It is also responsible for flexion and extension of the metacarpophalangeal and interphalangeal joints of the thumb and fingers. The radial nerve can be injured during surgical procedures involving the forearm or elbow, leading to specific motor deficits.

**Why the Correct Answer is Right:** In the given scenario, the patient has a fracture of the radial neck and post-operatively experiences inability to extend fingers at metacarpophalangeal joints, but no sensory loss in the hand. This suggests a motor deficit due to radial nerve injury. The radial nerve supplies motor function to the flexor muscles of the forearm and finger extensors, which are involved in extension.

**Why Each Wrong Option is Incorrect:**

A. The ulnar nerve (Option A) is responsible for the flexion of the fingers, which is not mentioned as a deficit in the question.
B. The median nerve (Option B) is responsible for sensation in the radial half of the dorsum of the hand and flexor muscles of the fingers, not extensor muscles involved in extension.
C. The musculocutaneous nerve (Option C) is responsible for the flexion of the lateral two-thirds of the forearm and the anterior surface of the arm, not extensor muscles involved in extension.
D. The posterior interosseous nerve (Option D) is responsible for the flexion of the wrist and fingers, not the extension deficit described.

**Clinical Pearl:** When assessing radial nerve injury, it is essential to consider the specific motor deficits described. In this case, the patient is unable to extend her fingers at the metacarpophalangeal and interphalangeal joints, which corresponds to radial nerve injury. This injury may occur during surgical procedures involving the radial neck or elbow, leading to specific motor deficits. Early recognition and appropriate management can improve the patient's prognosis and minimize permanent damage.