**Core Concept**
The radial nerve is a vital nerve in the upper limb, responsible for innervating muscles and providing sensory innervation to the skin. It originates from the posterior cord of the brachial plexus, which is formed from the C5-T1 nerve roots.

**Why the Correct Answer is Right**
The radial nerve is indeed a branch of the posterior cord of the brachial plexus, making **statement 1** true. It is also the nerve responsible for innervating the extensor compartment of the forearm, which includes muscles such as the extensor carpi radialis brevis and extensor digitorum, making **statement 2** true. The radial nerve arises from the C5-T1 nerve roots, which is a characteristic feature of its origin, making **statement 3** true. Additionally, the radial nerve provides sensory innervation to the skin of the extensor compartment of the forearm, making **statement 5** true.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because it incorrectly states that statement 4 is false, when in fact it is true. The anterior interosseous nerve is a branch of the median nerve, not the radial nerve.

**Option B:** This option is incorrect because it incorrectly states that statement 4 is true, when in fact it is false. The anterior interosseous nerve is a branch of the median nerve, not the radial nerve.

**Option D:** This option is incorrect because it states that all statements are true, when in fact statement 4 is false.

**Clinical Pearl / High-Yield Fact**
It's essential to remember the relationship between the radial nerve and the brachial plexus, as well as its innervation to the muscles and skin of the upper limb. The radial nerve is often affected in conditions such as radial tunnel syndrome, which can cause pain and weakness in the forearm and hand.

**✓ Correct Answer: C. 1,2,3,5 true & 4 false**