The coefficient of $x^5$ in ${\left(1+x\right)}^{21}+{\left(1+x\right)}^{22}+...+{\left(1+x\right)}^{30} is$

$\begin{array}{rcl} & =& {(1+\mathrm{x})}^{21}+{(1+\mathrm{x})}^{22}+.....+{(1+\mathrm{x})}^{30} \\ & =& {(1+\mathrm{x})}^{21} \left(1+(1+\mathrm{x})+{(1+\mathrm{x})}^2+....{(1+\mathrm{x})}^9\right) \\ & =& {(1+\mathrm{x})}^{21}, \frac{\left({\left(1+\mathrm{x}\right)}^{10}-1\right)}{(1+\mathrm{x})-1}\\ & =& \frac{\left[{\left(1+\mathrm{x}\right)}^{31}-{(1+\mathrm{x})}^{21}\right]}{\mathrm{x}}\end{array}$

Coefficient of  ${\mathrm{x}}^5=$Coeff ${\mathrm{x}}^6$ in Nr

               $={}^{31}{\mathrm{C}}_6-{}^{21}{\mathrm{C}}_6$