The basic strength of aliphatic amines in an aqueous solution is governed by a combination of factors: the +I (inductive) effect of the alkyl groups, the extent of hydrogen bonding (solvation) of the protonated amine, and steric hindrance.

For methyl-substituted amines:

1. +I Effect: Increases basicity as the number of methyl groups increases ($3^\circ > 2^\circ > 1^\circ$).
2. Solvation Effect: Decreases basicity as the number of methyl groups increases due to reduced hydrogen bonding ($1^\circ > 2^\circ > 3^\circ$).
3. Steric Hindrance: Bulky methyl groups hinder the approach of the proton and the solvent molecules.

The combined effect makes the secondary amine, dimethylamine ($(CH_3)_2NH$), the strongest base. Methylamine ($CH_3NH_2$) is stronger than trimethylamine ($(CH_3)_3N$) because the steric hindrance and poor solvation of the tertiary ammonium ion outweigh the +I effect. Therefore, the correct order is $(CH_3)_2NH > CH_3NH_2 > (CH_3)_3N$.