Localise the lesion by using the single most specific sign first - the nystagmus.
Map the cerebellum onto its three functional zones:
• Vestibulocerebellum = flocculonodular lobe → equilibrium, balance and the control of eye movements (so its damage → nystagmus + truncal/gait imbalance).
• Spinocerebellum = vermis + anterior lobe → posture and gait coordination.
• Cerebrocerebellum = lateral hemispheres (output via the dentate nucleus) → planning of fine, skilled limb movements.
This patient after head trauma shows a drunken, wide-based ("alcoholic") gait together with nystagmus. Of all the listed regions, the one whose function explicitly couples balance with eye-movement control is the flocculonodular lobe - because it is the cerebellum's interface with the vestibular system, its injury characteristically produces both unsteady gait and nystagmus.
Cross-checking the others: the dentate is a deep nucleus (its lesions give intention tremor and limb dysmetria, not nystagmus); the anterior lobe is the classic chronic-alcoholic site causing gait ataxia but is not the eye-movement centre; and the vermis produces truncal ataxia without nystagmus being its hallmark. The pairing of nystagmus with gait disturbance is what selects the flocculonodular lobe.
Therefore option A is correct.