Step 1: Understanding the Concept
For amines of similar molar mass, boiling point depends on intermolecular hydrogen bonding through N-H bonds.
Step 2: Compare the types
A primary amine has two N-H bonds and forms the most hydrogen bonds.
A secondary amine has one N-H bond and forms fewer.
A tertiary amine has no N-H bond, so it cannot hydrogen bond with its own molecules.
Step 3: Order
Boiling point order is primary > secondary > tertiary, that is (III) > (II) > (I).
Final Answer:
The order is primary > secondary > tertiary.
\[ \boxed{\text{(D) }(\text{III}) > (\text{II}) > (\text{I})} \]