The correct order of increasing intermolecular hydrogen bond strength is

🧠 H-bond strength depends on the electronegativity of the atom bonded to H and the polarity of the X–H bond (donor strength).

🗺️ Analyse each molecule:

• HCN ($\mathrm{H-C\equiv N}$): H is bonded to C (low electronegativity, poor H-bond donor); only weak interaction possible → weakest ✗
• $\mathrm{H_2O}$: H bonded to O; O is highly electronegative → strong O–H···O H-bond; O also has 2 lone pairs as acceptors
• $\mathrm{NH_3}$: H bonded to N; N is electronegative; N has a concentrated lone pair → forms N–H···N H-bonds

⚠️ $\mathrm{CH_4}$ forms no H-bonds at all (H bonded to C, which is non-electronegative).

Increasing H-bond strength:

$\mathrm{HCN < H_2O < NH_3}$

⚡ Among these three, HCN is the weakest donor (C–H bond). Between $\mathrm{H_2O}$ and $\mathrm{NH_3}$, while O is more electronegative, the lone pair on N in $\mathrm{NH_3}$ is more available and directional, contributing to stronger effective H-bonding in this comparison.

$\boxed{\text{Answer: HCN} < \mathrm{H_2O} < \mathrm{NH_3}}$