Step 1: Understanding the Concept:
Basic strength in aqueous solution depends on three factors: Inductive effect ($+I$), Solvation effect (Hydrogen bonding with water), and Steric hindrance.
Step 2: Formula Application:
For Ethyl substituted amines ($C_2H_5$), the secondary amine is always the most basic. The order is:
$2^{\circ}>3^{\circ}>1^{\circ}>NH_3$.
Step 3: Explanation:
In ethyl amines, the $+I$ effect and solvation effect balance out such that the $2^{\circ}$ amine is strongest. Unlike methyl amines (where $2^{\circ}>1^{\circ}>3^{\circ}$), the larger ethyl group makes the $3^{\circ}$ amine more basic than the $1^{\circ}$ amine.
Step 4: Final Answer:
The order is $(C_2H_5)_2NH>(C_2H_5)_3N>C_2H_5NH_2>NH_3$.