According to the IUPAC nomenclature rules for coordination compounds:

1. Ligands are named in alphabetical order before the name of the central metal atom/ion.
2. The ligands present in the complex are: 'ammine' for $NH_3$, 'bromo' for $Br^-$, 'chloro' for $Cl^-$, and 'nitro' for $NO_2^-$.
3. Arranging them alphabetically gives: ammine, bromo, chloro, nitro. The prefix 'tri' is used to indicate three ammine ligands, making it 'triammine'.
4. Since the coordination sphere is a neutral complex molecule, the central metal is simply named 'cobalt' (the suffix '-ate' is used only for anionic complexes).
5. The oxidation state of cobalt (let it be $x$) is calculated as: $x + 3(0) + (-1) + (-1) + (-1) = 0 \implies x = +3$. This is indicated in Roman numerals as (III). Combining all these parts, the correct IUPAC name is Triamminebromochloronitrocobalt(III).