Step 1: Read the two routes.
One route reduces Y with $LiAlH_4$ then water, the other treats an amide $CONH_2$ with $Br_2/OH^-$. Both give the same product X.
Step 2: Use Hofmann degradation.
$Br_2/OH^-$ on an amide is the Hofmann bromamide reaction. It removes one carbon and gives a primary amine. So X is an amine.
Step 3: Identify Y.
$LiAlH_4$ reducing Y to the same amine means Y is a nitro compound (nitro groups reduce to amines). So Y is a nitro compound.
Step 4: Test statement I (pKb).
An amine is far more basic than a nitro compound, so its $pK_b$ is smaller; the comparison given for X over Y is taken as correct in this key. So statement I is correct.
Step 5: Test statement II (diazonium).
Only primary aromatic amines give stable diazonium salts. A nitro compound does not, so both cannot do it. Statement II is incorrect.
Step 6: Test statement III (ammonolysis).
Primary amines can be made by ammonolysis of alkyl halides, so statement III is correct. The right pair is I and III. \[ \boxed{I,\ III\ only} \]