The correct order of increasing the basic nature of the bases ammonia, methylamine, and ethylamine in the gas phase is :
A. \[{\left( {{\rm{C}}{{\rm{H}}_{\rm{3}}}} \right)_{\rm{2}}}{\rm{NH < N}}{{\rm{H}}_{\rm{3}}}{\rm{ < C}}{{\rm{H}}_{\rm{3}}}{\rm{N}}{{\rm{H}}_{\rm{2}}}\]
B. \[{\rm{N}}{{\rm{H}}_{\rm{3}}}{\rm{ < C}}{{\rm{H}}_{\rm{3}}}{\rm{N}}{{\rm{H}}_{\rm{2}}}{\rm{ < }}{\left( {{\rm{C}}{{\rm{H}}_{\rm{3}}}} \right)_{\rm{2}}}{\rm{NH}}\]
C. \[{\rm{C}}{{\rm{H}}_{\rm{3}}}{\rm{N}}{{\rm{H}}_{\rm{2}}}{\rm{ < }}{\left( {{\rm{C}}{{\rm{H}}_{\rm{3}}}} \right)_{\rm{2}}}{\rm{NH < N}}{{\rm{H}}_{\rm{3}}}\]
D. \[{\rm{C}}{{\rm{H}}_{\rm{3}}}{\rm{N}}{{\rm{H}}_{\rm{2}}}{\rm{ < N}}{{\rm{H}}_{\rm{3}}}{\rm{ < }}{\left( {{\rm{C}}{{\rm{H}}_{\rm{3}}}} \right)_{\rm{2}}}{\rm{NH}}\]

Hint: Amines are organic compounds developed by substituting one or more hydrogen atoms with alkyl or aryl groups. The nitrogen atom in amine has a lone pair of electrons, which is why it is basic.

Complete Step by Step Solution:
Lewis's concept states that a base is a substance that can give a pair of electrons to another substance.
The nitrogen atom in amine has a lone pair of electrons. So, amines are basic.
The basicity of amines is due to their structure.
Amines provide a pair of electrons to another substance forming an ammonium cation.
The more stable this cation, the more will be its basicity.
The basicity of amines in the gaseous phase is due to the +I-effect of the alkyl groups.
Dimethyl amine
It loses a pair of electrons and forms an ammonium cation.

Image: Cation formed by dimethylamine.
Alkyl groups are the electron releasing group or +I group.
Here, two methyl groups are present, due to this the positive charge present on the cation gets dispersed. This increases the stability of the cation.

Methyl amine
It loses a pair of electrons and forms an ammonium cation.

Image: Cation formed by methylamine.
Alkyl groups are the electron releasing group or +I group.
Here one methyl group is present, due to this the positive charge present on the cation gets dispersed. This increases the stability of the cation. But methyl amine is less basic than Dimethylamine.

Ammonia


Image: Cation formed by ammonia.
Here no methyl groups are present. The positive charge on the carbon is not stabilised.
So, ammonia is the least basic out of the given options.
So, the basicity of amines should be in the order
\[{\rm{N}}{{\rm{H}}_{\rm{3}}}{\rm{ < C}}{{\rm{H}}_{\rm{3}}}{\rm{N}}{{\rm{H}}_{\rm{2}}}{\rm{ < }}{\left( {{\rm{C}}{{\rm{H}}_{\rm{3}}}} \right)_{\rm{2}}}{\rm{NH}}\]
So, option B is correct.

Note: Dimethyl amine is more basic than methyl amine and ammonia as the cation formed by Dimethyl amine is most stable. Two methyl groups present help in positive charge delocalization. There is one methyl group present in methyl amine due to which its cation is stable but not as stable as in the case of ethyl amine. Ammonia is the least stable as it has no presence of the methyl group.