Concept:
The basic character of amines depends upon the availability of the lone pair of electrons on nitrogen for donation to a proton. In aqueous solution, not only the electron-releasing inductive effect of alkyl groups but also the solvation of the conjugate acid plays a very important role.
Step 1: Consider the effect of alkyl groups.
Methyl groups exhibit a positive inductive effect $(+I)$ and increase the electron density on nitrogen.
Therefore,
\[
(\mathrm{CH_3})_3\mathrm{N}
>
(\mathrm{CH_3})_2\mathrm{NH}
>
\mathrm{CH_3NH_2}
>
\mathrm{NH_3}
\]
would be expected if only the inductive effect were considered.
Step 2: Consider solvation in aqueous medium.
The protonated amines are stabilized by hydrogen bonding with water molecules.
The extent of solvation follows:
\[
\mathrm{RNH_3^+}
>
\mathrm{R_2NH_2^+}
>
\mathrm{R_3NH^+}
\]
because tertiary ammonium ions are sterically hindered and are less effectively solvated.
Step 3: Combine both effects.
In aqueous solution, secondary amines are the strongest bases because they receive sufficient electron donation from alkyl groups and are still reasonably solvated.
Thus,
\[
(\mathrm{CH_3})_2\mathrm{NH}
>
\mathrm{CH_3NH_2}
>
(\mathrm{CH_3})_3\mathrm{N}
>
\mathrm{NH_3}
\]
in decreasing order of basic strength.
Step 4: Write increasing order.
\[
\boxed{
\mathrm{NH_3}
<
(\mathrm{CH_3})_3\mathrm{N}
<
\mathrm{CH_3NH_2}
<
(\mathrm{CH_3})_2\mathrm{NH}
}
\]
Hence,
\[
\boxed{\text{Option (1)}}
\]