n- Butylamine (I), diethylamine (II) and N, N-dimethylethyl amine (III) have the same molar mass. The increasing order of their boiling point is

n - Butylamine is a primary amine and it can form 2 hydrogen bonds.

diethylamine is a secondary amine and it can form 1 hydrogen bond.

N, N-dimethylethyl amine is a tertiary amine and it cannot form any hydrogen bond.

Hydrogen bonding $\alpha$ Boiling point of a compound

Thus, the order of tendency to form hydrogen bonding is $\mathrm{III}<\mathrm{II}<\mathrm{I}$.

Therefore, the order of boiling point is $\mathrm{III}<\mathrm{II}<\mathrm{I}$.

Hence, option (A) is correct.