Step 1: Recall the functional divisions of the cerebellum.
The cerebellum is divided into: the flocculonodular lobe (vestibulocerebellum) - controls balance/equilibrium and eye movements; the anterior lobe / vermis (spinocerebellum) - controls posture, gait and trunk/limb coordination; and the lateral hemispheres (cerebrocerebellum) - plan skilled, learned movements. The dentate is a deep nucleus, not a lobe.
Step 2: Match the two clinical signs to a region.
The patient has BOTH a wide-based ataxic ("alcoholic"/drunken) gait AND nystagmus. Nystagmus is the key localising sign: abnormal eye movements and disturbed equilibrium point specifically to the flocculonodular lobe (vestibulocerebellum), which links the cerebellum with the vestibular system controlling balance and gaze. Combined with truncal/gait instability, the flocculonodular lobe best explains both features together.
Step 3: Eliminate the distractors.
(B) Dentate nucleus is a deep cerebellar nucleus (not a "lobe") and lesions cause limb dysmetria/intention tremor, not prominent nystagmus. (C) Anterior lobe damage (classic in chronic alcoholics) gives gait/trunk ataxia but is not the lobe most associated with nystagmus. (D) Vermis lesions cause truncal ataxia and a broad-based gait but, again, nystagmus is the hallmark of the flocculonodular/vestibulocerebellar region. Because the question pairs gait disturbance WITH nystagmus, the flocculonodular lobe is the best single answer.
Final answer: A - Flocculonodular lobe.