The order of basic strength of amine in aqueous solution is:

\(C_6H_5CH_2NH_2 < C_2H_5NH_2 < (C_2H_5)_3N  < (C_2H_5)_2NH\)

The correct answer is option 1. \(C_6H_5CH_2NH_2 < C_2H_5NH_2 < (C_2H_5)_3N  < (C_2H_5)_2NH\).

Let us delve into the explanation:

A. \(C_6H_5CH_2NH_2\): This compound is benzylamine, which contains a benzyl group attached to the amino group. The benzyl group can donate electrons through resonance, which delocalizes the lone pair on the nitrogen atom, making it less available for protonation. Therefore, benzylamine is expected to be the weakest base among the given options.

2. \(C_2H_5NH_2\): Ethylamine is a primary amine with only one ethyl group attached to the nitrogen atom. The lone pair on nitrogen is relatively more available compared to secondary and tertiary amines. Therefore, ethylamine is expected to be stronger than benzylamine but weaker than the remaining options.

3. \((C_2H_5)_3N\): Triethylamine is a tertiary amine with three ethyl groups attached to the nitrogen atom. Tertiary amines are generally stronger bases than primary and secondary amines because the electron-donating groups stabilize the resulting ammonium ion. But the tertiary amines are found to be  less basic than secondary amines. The reason is that a tertiary amine, though has three alkyl groups which can donate electrons to the nitrogen atom but they also cause crowding (also called steric hinderance) around nitrogen. This hinders the protonation at nitrogen atom and hence, reduces the basicity. Therefore, triethylamine is less basic then diethylamine.

4. \((C_2H_5)_2NH\): Diethylamine is a secondary amine with two ethyl groups attached to the nitrogen atom. Secondary amines typically fall between primary and tertiary amines in terms of basic strength. Therefore, diethylamine is expected to be stronger than ethylamine but weaker than triethylamine.

Considering these factors, the correct order of basic strength is indeed: Option 1: \(C_6H_5CH_2NH_2 < C_2H_5NH_2 < (C_2H_5)_3N < (C_2H_5)_2NH\)

This arrangement aligns with the expected trends based on the structures and electronic properties of the amines involved.