Step 1: Identify the given compounds.
\[
(a)=(C_2H_5)_2NH
\]
Diethylamine (secondary amine)
\[
(b)=C_2H_5NH_2
\]
Ethylamine (primary amine)
\[
(c)=NH_3
\]
Ammonia
\[
(d)=C_6H_5NH_2
\]
Aniline
Step 2: Recall the factors affecting basicity.
Basicity depends on the availability of the lone pair of electrons on nitrogen for donation.
Alkyl groups show a \(+I\) (electron-releasing) effect, which increases electron density on nitrogen and increases basicity.
Therefore,
\[
\text{Secondary amine}\gt \text{Primary amine}\gt \text{Ammonia}
\]
Step 3: Compare diethylamine, ethylamine and ammonia.
Diethylamine contains two ethyl groups that donate electron density to nitrogen through the \(+I\) effect.
Hence, its lone pair is most available.
\[
(C_2H_5)_2NH \gt C_2H_5NH_2 \gt NH_3
\]
Step 4: Compare aniline with ammonia.
In aniline, the lone pair on nitrogen is involved in resonance with the benzene ring.
\[
\text{Lone pair}
\longrightarrow
\text{delocalized into the aromatic ring}
\]
As a result, the lone pair becomes less available for protonation.
Therefore, aniline is less basic than ammonia.
\[
NH_3 \gt C_6H_5NH_2
\]
Step 5: Write the overall order.
Combining all comparisons:
\[
(C_2H_5)_2NH
\gt
C_2H_5NH_2
\gt
NH_3
\gt
C_6H_5NH_2
\]
That is,
\[
a\gt b\gt c\gt d
\]
Step 6: Final conclusion.
Hence, the correct order of basicity is
\[
\boxed{a\gt b\gt c\gt d}
\]
Therefore, option (2) is correct.