After acylation the formula will be $\mathrm{R}-\left({\mathrm{NHCOCH}}_3\right)\mathrm{x}$. Hence, the change in the molecular mass is calculated as below

$$\begin{gathered} For\mathit{ }\mathit{1}one\mathit{ }N{H}_{\mathit{2}}\mathit{\to }\mathit{14}\mathit{+}\mathit{2}\mathit{=}\mathit{16}\mathit{ }units \\ When\mathit{ }It\mathit{ }turns\mathit{ }into\mathit{ }NH\mathit{-}COC{H}_{\mathit{3}}\mathit{ }\mathit{ }\mathit{}\mathit{ }\mathit{ }\mathit{\to }\mathit{14}\mathit{+}\mathit{1}\mathit{+}\mathit{12}\mathit{+}\mathit{16}\mathit{+}\mathit{12}\mathit{+}\mathit{3}\mathit{=}\mathit{58}\mathit{ } \\ Change\mathit{ }in\mathit{ }mass\mathit{\Rightarrow }\mathit{58}\mathit{-}\mathit{16}\mathit{=}\mathit{42}units\mathit{.} \\ \mathit{ }So\mathit{ }for\mathit{ }every\mathit{ }N{H}_{\mathit{2}}\mathit{ }\mathit{,}\mathit{ }there\mathit{ }will\mathit{ }be\mathit{ }an\mathit{ }increase\mathit{ }of\mathit{ }\mathit{42}g\mathit{ }after\mathit{ }acylation\mathit{.}\mathit{ }\mathit{ } \\ \mathit{\Rightarrow }\mathit{ }The\mathit{ }total\mathit{ }increase\mathit{ }in\mathit{ }mass\mathit{ }\mathit{=}\mathit{ }\mathit{390}\mathit{-}\mathit{180}\mathit{ }\mathit{=}\mathit{ }\mathit{210}\mathit{ }\mathit{ } \\ \mathit{\Rightarrow }\mathit{ }No\mathit{.}\mathit{ }of\mathit{ }amino\mathit{ }groups\mathit{=}\frac{\mathit{210}}{\mathit{42}}\mathit{=}\mathit{ }\mathit{5}\mathit{ } \\ Hence\mathit{ }number\mathit{ }of\mathit{ }amino\mathit{ }groups\mathit{ }are\mathit{ }\mathit{5}\mathit{.} \end{gathered}$$

$(12+16+12+3-1)\mathrm{x}=390-180$

$\mathrm{or} 42\mathrm{x}=210; \mathrm{or} \mathrm{x}=5$