The coefficient of ${\mathrm{x}}^{50}$ in the expansion of $(1+\mathrm{x}{)}^{101}(1-\mathrm{x}+{\mathrm{x}}^2{)}^{100}$ is

$=(1+x{)}^{101}{\left(1-x+x^2\right)}^{100}$

$=(1+x)\left[(1+x{)}^{100}{\left(1-x+x^2\right)}^{100}\right]$

$=(1+x){\left(1-x^3\right)}^{100}$

$={\left(1-x^3\right)}^{100}+x{\left(1-x^3\right)}^{100}$

$={\left(1-x^3\right)}^{100}+x{\left(1-x^3\right)}^{100}$

The coefficient of ${\mathrm{x}}^{50}$ in the expansion of ${\left(1-x^3\right)}^{100}+x{\left(1-x^3\right)}^{100}$

The coefficient of ${\mathrm{x}}^{50}$ in ${\left(1-x^3\right)}^{100}$ +The coefficient of ${\mathrm{x}}^{49}$ in  $x{\left(1-x^3\right)}^{100}$

=0