The number of terms in the expansion of ${\left(1+\mathrm{x}\right)}^{101}{\left(1+{\mathrm{x}}^2-\mathrm{x}\right)}^{100}$ in powers of x

$$\begin{gathered} {\left(1+\mathrm{x}\right)}^{101} {\left(1-\mathrm{x}+{\mathrm{x}}^2\right)}^{100} \\ =(1+\mathrm{x}){(1+\mathrm{x})}^{100}{(1-\mathrm{x}+{\mathrm{x}}^2)}^{100} \\ =(1+\mathrm{x}){(1+{\mathrm{x}}^3)}^{100} \\ ={(1+{\mathrm{x}}^3)}^{100}+\mathrm{x}{(1+{\mathrm{x}}^3)}^{100} \end{gathered}$$

number of terms in ${(1+{\mathrm{x}}^3)}^{100}=101$ 

Since there is no common terms 

$\left(1+\mathrm{x}\right){\left(1+{\mathrm{x}}^3\right)}^{100}=101+101=202$