What is the chemical name of the compound ${\mathrm{FeCl}}_3$?

The correct option is C Iron(III) chloride
Iron shows variable valencies.

$\bullet$ When it shows valency - 2, it is represented as ferrous or iron(II).

$\bullet$ When it shows valency - 3, it is represented as ferric or iron(III).

The valency of chloride ion is 1, hence in ${\text{FeCl}}_3$, the valency of iron is 3. So, the name of the compound is iron(III) chloride or ferric chloride.