Brachial plexus
The nerve root of the axillary nerve is C5 and C6 (choice A). Injury to the axillary nerve may occur in patients with anterior dislocation of a shoulder after falling on an outstretched hand. Injury to the axillary nerve presents with flattening of the deltoid muscle after injury, loss of lateral rotation and abduction of the affected shoulder due to deltoid muscle weakness and loss of sensation over the lateral aspect of arm.
Reference: First Aid for the USMLE Step 1 (2016): pp. 419, 423
C5, C6 and C7 of the lateral cord (choice B) are the nerve roots of the musculocutaneous nerve. Injury to this nerve causes loss of elbow flexion, weakness in supination and sensation loss on the lateral aspect of the forearm.
C5, C6, C7, C8 and T1 of the lateral and medial cords (choice C) are the nerve roots of the median nerve. Median nerve injury is common in supracondylar fracture of humerus. Injury to the nerve causes loss of flexion of digits, thenar muscle and lumbricals 1 and 2, weakness of wrist flexion, loss of pronation and sensory loss on lateral palm and 3 half digits.
C5, C6, C7, C8 and T1 of the posterior cord (choice D) are the nerve roots of the radial nerve. Injury to the radial nerve commonly occurs in a midshaft fracture of humerus, resulting in wrist drop.
C8 and T1 of the medial cord (choice E) are the nerve roots of the ulnar nerve. Injury to the ulnar nerve causes claw hand. Injury of the ulnar nerve will cause weakened flexion of digits 4 and 5 and sensory loss on digits 5 and one half of 4.
The axillary nerve (choice A) originates from the posterior cord of the brachial plexus and carries fibers from C5 and C6. The axillary nerve supplies the deltoid (a muscle of the shoulder), teres minor (one of the rotator cuff muscles), and triceps brachii (long head). The axillary nerve also carries sensory fibers from the shoulder joint, as well as the skin covering the inferior region of the deltoid muscle. It may be injured in dislocations of the shoulder joint, compression of the axilla with a crutch or fracture of the surgical neck of the humerus.
The median nerve (choice B) is formed by fibers from the lateral and medial cords of the brachial plexus, originating from ventral roots of C5, C6 and C7 (lateral cord) and C8 and T1 (medial cord). It innervates all of the flexors in the forearm except flexor carpi ulnaris and that part of flexor digitorum profundus that supplies the medial two digits. In the hand, the median nerve supplies motor innervation to the 1st and 2nd lumbrical muscles. It also supplies the muscles of the thenar eminence by a recurrent thenar branch. The median nerve innervates the skin of the palmar side of the thumb, the index and middle finger and half the ring finger. It is the only nerve that passes through the carpal tunnel where it is commonly compressed. The lateral part of the palm is supplied by the palmar cutaneous branch of the median nerve, which leaves the nerve proximal to the wrist creases and so is not involved in carpal tunnel syndrome.
The musculocutaneous nerve (choice C) arises from the lateral cord of the brachial plexus, opposite the lower border of the pectoralis major, its fibers being derived from C5, C6 and C7. The nerve usually passes through the coracobrachialis and between the biceps and the brachialis and at the elbow it becomes the lateral antebrachial cutaneous nerve. Above the elbow it supplies motor nerves to coracobrachialis, biceps, and the greater part of the brachialis. Below the elbow it contains only sensory fibers. Damage to the shoulder and brachial plexus or compression by the biceps aponeurosis and tendon can affect the musculocutaneous nerve. Lesions produce weakness of flexion at the elbow and weakness of supination. The biceps is an important supinator. There is sensory loss on the lateral side of the forearm.
The radial nerve (choice D) may be damaged by trauma or entrapment in the axilla, the upper arm, the elbow or the wrist. In the spiral groove of the humerus, it may be injured with a fracture of the shaft of the humerus. A lesion at this level will result in wrist drop.
The radial nerve is the terminal branch of the posterior cord of the brachial plexus and carries fibers from C5-C8 and a sensory component from T1. The nerve and its branches provide motor innervation to the dorsal arm muscles (the triceps brachii and the anconeus) and the extrinsic extensors of the wrists and hands. It also provides sensory innervation to most of the back of the hand. Compression injuries may occur in the axilla, possibly with medial and ulnar nerve involvement as well. Saturday night syndrome is well known. It is so named because it can be acquired by sleeping with the arm over the back of a chair while in a drunken state, so compressing the brachial plexus (it can also be caused by using crutches). Fractures or dislocation of the head of the humerus may also damage the nerve in the axilla. Fracture is the usual cause of damage in the upper arm as in this case. Entrapment of the radial nerve (choice D) can occur at the elbow. Fractures or compression may damage the nerve at the wrist producing a finger drop with normal wrist movement. Lesions from compression or simple fractures usually recover spontaneously.
Reference: First Aid for the USMLE Step 1 (2016): pp. 419
The ulnar nerve (choice E) originates from the C8-T1 nerve roots, which form part of the medial cord of the brachial plexus. The nerve runs near the ulna bone. It is the largest unprotected nerve in the body (meaning unprotected by muscle or bone), so injury is common particularly at the elbow near the medial epicondyle. It supplies the little finger, and the adjacent half of the ring finger. One common cause is cubital tunnel syndrome. In this syndrome, a tunnel on the medial side of the elbow traps the nerve. Pinching of the ulnar nerve often causes paresthesia in the fourth and fifth digits. Temporary paresthesias can be caused by sleeping or poor posture while awkwardly placing weight on one's arm, or by cycling for extended periods of time.

Radial Nerve Palsy
Present with both motor and sensory deficits along the radial nerve distribution, mainly extension of the wrist and sensation to the dorsal aspects of the lateral 3 1/2 digits.
Radial nerve palsy is most commonly caused by fracture of the humerus, especially in the middle third (Holstein-Lewis fracture) or at the junction of the middle and distal thirds. The nerve injury may occur acutely at the time of injury, secondary to fracture manipulation, or chronically from a healing callus.