$\text{ If the coefficient of }{x}^{100}\text{ in }1+(1+x)+(1+x{)}^2+\dots +(1+x{)}^n;(n\ge 100){\text{ is }}^{201}{C}_{101}\text{ , then }n\text{ is equal to }$

 

 

$$\begin{gathered} 1+(1+x)+(1+x{)}^2+\dots +(1+x{)}^n \\ =\frac{(1+x{)}^{n+1}-1}{1+x-1}=\frac{(1+x{)}^{n+1}-1}{x} \\ \mathrm{the} \mathrm{coefficient}\mathrm{of} x^{100} \mathrm{in} \frac{(1+x{)}^{n+1}-1}{x}=\mathrm{the} \mathrm{coefficientof} x^{101} \mathrm{in} (1+x{)}^{n+1}-1{=}^{n+1}{C}_{101}{=}^{201}{C}_{101} \\ \Rightarrow n=200 \end{gathered}$$