Step 1: Localise before you name.
Rather than test each eponym blindly, first place the lesion anatomically using the three deficits, then read off the matching syndrome.
Step 2: Interpret each sign.
• Left arm and leg weakness (contralateral hemiparesis): the corticospinal tract is hit on the right, above the medullary decussation.
• Right facial paralysis (ipsilateral, lower-motor-neuron type): the facial nerve nucleus / fascicle (CN VII) is involved on the right.
• Difficulty with horizontal eye movements: points to the abducens (CN VI) region - CN VI and VII nuclei sit together in the pons.
Step 3: Combine the localisation.
A right ventral/caudal pontine lesion that catches the corticospinal tract (crossed limb weakness) plus the ipsilateral CN VI and CN VII fibres gives the classic ventral pontine (Millard-Gubler / Raymond-Cestan-type) crossed picture: contralateral hemiplegia with ipsilateral facial and lateral-gaze palsy. This is Millard-Gubler syndrome.
Step 4: Rule out the others by their level.
• Benedikt - midbrain; ipsilateral CN III palsy with contralateral tremor/involuntary movements (red nucleus). Wrong level.
• Wallenberg (lateral medullary) - vertigo, dysphagia, crossed sensory loss, Horner's; no facial LMN palsy or pyramidal limb weakness as described.
• Foville - also dorsal pontine, but features a conjugate horizontal gaze palsy toward the lesion; the described picture of discrete VI/VII plus crossed hemiplegia fits Millard-Gubler best.
Step 5: Conclude.
The ventral pontine crossed syndrome is Millard-Gubler.
Final Answer: Option 3 - Millard-Gubler syndrome.