The formula for manganese(III) oxide is Mn₂O₃. Here's how we arrive at that:

* Manganese(III) indicates that manganese has a +3 charge (Mn³⁺).

* Oxide refers to oxygen, which typically has a -2 charge (O²⁻).

* To balance the charges, we need two manganese ions (2 x +3 = +6) and three oxygen ions (3 x -2 = -6).

Therefore, the formula is Mn₂O₃.