Correct order of basic strength of given amines is:
(A) \[M{e_2}NH > MeN{H_2} > M{e_3}N > N{H_3}\](protic solvent)
(B) \[E{t_2}NH > E{t_3}N > EtN{H_2} > N{H_3}\](protic solvent)
(C) \[M{e_3}N > M{e_2}NH > MeN{H_2} > N{H_3}\] (gas phase)
(D) All are correct

Hint: Try to recall that basicity of amines can be determined with help of factors like steric factors, hydrogen bonding and electronegativity. Now, by using this you can easily find the correct option from the given options.

Complete answer:
* It is known to you that amines are basic in nature as they have a lone pair of electrons on nitrogen. Therefore, they have a strong tendency to donate the lone pair of electrons to electron acceptors.
* The basicity of amines depends on:
The electronic properties of the substituent’s (alkyl groups enhance the basicity, aryl groups diminish it) and also on the degree of salvation of the protonated amine, which includes steric hindrance by the groups on nitrogen.
* In the gaseous phase, for R=Me, the order of basic strength is \[M{e_3}N > M{e_2}NH > MeN{H_2} > N{H_3}\] but in protic solvent like water, the order of basic strength is \[M{e_2}NH > MeN{H_2} > M{e_3}N > N{H_3}\]
* This happens because in the gaseous phase, basicity gets stronger due to extra electron donating (+I) alkyl groups but in protic solvent another effect is competing which is solvent effect.
* According to the solvent effect the ammonium salts in solution are stabilized not only by alkyl groups, but also by hydrogen-bond donation.
* In protic solvent like water, for R= Et, the order of basic strength is: \[E{t_2}NH > E{t_3}N > EtN{H_2} > N{H_3}\]. This happens because of the steric factor of Et.
Therefore, from above we can conclude that option D is the correct option to the given question.
Note:
It should be remembered to you that aliphatic amines are more basic compared to aromatic amines. This happens because the lone pair present on nitrogen atoms is delocalised. So, the availability of the lone pair of electrons will be less and hence will be less basic than aliphatic amines.