The answer is: the coefficients are 2, 3, 3, 2.
Balanced chemical reaction: 2FeCl₃ + 3Zn → 3ZnCl₂ + 2Fe.
Oxidation reaction: Zn⁰ → Zn⁺² + 2e⁻ /×3.
Reduction reaction: Fe⁺³ + 3e⁻ → Fe⁰ /×2.
Oxidation reaction: 3Zn⁰ → 3Zn⁺² + 6e⁻.
Reduction reaction: 2Fe⁺³ + 6e⁻ → 2Fe⁰.
Zinc change oxidation number from 0 to +2 (oxidation) and iron change oxidation number from +3 to 0 (reduction).