**Core Concept**
Haemophilus influenzae is a Gram-negative bacterium that can cause meningitis, a life-threatening infection of the protective membranes covering the brain and spinal cord. The production of beta-lactamase enzymes by the bacteria renders them resistant to beta-lactam antibiotics, such as penicillins and cephalosporins. Additionally, chloramphenicol resistance is a concern, making it essential to select an antimicrobial agent that is effective against beta-lactamase-producing H. influenzae.

**Why the Correct Answer is Right**
The correct choice is a third-generation cephalosporin, such as cefotaxime or ceftriaxone. These antibiotics have a broad spectrum of activity against Gram-negative bacteria, including H. influenzae, and are resistant to beta-lactamase degradation. Ceftriaxone, in particular, has a long half-life, allowing for once-daily dosing and making it an ideal choice for treating meningitis. The efficacy of ceftriaxone against chloramphenicol-resistant H. influenzae further supports its selection in this scenario.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because ampicillin, a penicillin antibiotic, is typically ineffective against beta-lactamase-producing H. influenzae due to the enzyme's ability to degrade the beta-lactam ring.

**Option B:** This option is incorrect because chloramphenicol, although effective against many Gram-negative bacteria, is not the best choice in this scenario due to the strain's resistance to this antibiotic.

**Option C:** This option is incorrect because trimethoprim-sulfamethoxazole (TMP-SMX) has limited activity against H. influenzae and is not the preferred treatment for meningitis caused by this organism.

**Clinical Pearl / High-Yield Fact**
When treating bacterial meningitis, it is essential to choose an antimicrobial agent that is effective against the causative organism and has a good penetration into the cerebrospinal fluid (CSF). In the case of H. influenzae meningitis, a third-generation cephalosporin, such as ceftriaxone, is the preferred choice due to its broad spectrum of activity, resistance to beta-lactamase degradation, and excellent CSF penetration.

**Correct Answer:** C. Ceftriaxone