Which of the following is least basic?

The correct option is A

${\left({\mathrm{CH}}_3\mathrm{CO}\right)}_2\mathrm{NH}$

The explanation for the correct option:

(a) ${\left({\mathrm{CH}}_3\mathrm{CO}\right)}_2\mathrm{NH}$

1. Basicity of amines is the tendency to donate the lone pair of electrons of $\mathrm{N}$(nitrogen) in a chemical reaction, more easily it can donate more will be the basicity.
2. The presence of electron-withdrawing and conjugation of lone pair decreases the basicity.
3. The presence of an electron-donating group increases the basicity.
4. Here, the lone pair of $\mathrm{N}$ is engaged in conjugation with two carbonyl groups and the lone pair is not available for donation conjugation.
5. So, it will be the least basic.

The explanation for incorrect option:

(b) $\left({\mathrm{CH}}_3\mathrm{CO}\right){\mathrm{NHC}}_2{\mathrm{H}}_5$

1. Here, there is carbonyl group which will be engage the lone pair in conjugation.
2. But, there is alkyl group as well, which shows +I effect.
3. So, it will be more basic than (a)

(c) ${\left({\mathrm{C}}_2{\mathrm{H}}_5\right)}_3\mathrm{N}$

1. It is a tertiary amine with three alkyl group which will show +I effect and increase the basicity.
2. It will be most basic.

(d) ${\left({\mathrm{C}}_2{\mathrm{H}}_5\right)}_2\mathrm{NH}$

1. It is a secondary amine with two alkyl which will show +I effect and increase the basicity.
2. It will be more basic but less than (c)

Therefore, Option (a) is correct option, ${\left({\mathrm{CH}}_3\mathrm{CO}\right)}_2\mathrm{NH}$ is least basic.