Identify the correct IUPAC name for the given organic compound.

According to IUPAC nomenclature rules, the principal functional group is the ketone ($-C=O$), which must receive the lowest possible locant.

The longest carbon chain containing the ketone group has $7$ carbon atoms, so the parent chain is heptane.

Numbering the chain from left to right places the ketone at position $3$. Numbering from right to left places it at position $5$. Therefore, the correct numbering is from left to right.

The substituents and their positions are:
- A chloro group at C-$2$
- A methyl group at C-$4$
- A bromo group at C-$6$

The substituents must be listed in alphabetical order: bromo, chloro, methyl.

Combining these parts, the correct IUPAC name is $6$-bromo-$2$-chloro-$4$-methylheptan-$3$-one.

Answer: $6$-bromo-$2$-chloro-$4$-methylheptan-$3$-one