The coefficient of $x^5$ in the expansion of ${(1+x)}^{21}+{(1+x)}^{22}+{(1+x)}^{23} +........... + {(1+x)}^{30}$ is

$(1+\mathrm{x}{)}^{21}+(1+\mathrm{x}{)}^{22}+\dots +(1+\mathrm{x}{)}^{30}$
$=(1+\mathrm{x}{)}^{21}\left[\frac{(1+\mathrm{x}{)}^{10}-1}{(1+\mathrm{x})-1}\right]=\frac{1}{\mathrm{x}}\left[(1+\mathrm{x}{)}^{31}-(1+\mathrm{x}{)}^{21}\right]$
$$\begin{gathered} \Rightarrow \mathrm{Coefficient} \mathrm{of} {\mathrm{x}}^5 \mathrm{in} \mathrm{the} \mathrm{given} \mathrm{expression} \\ =\text{ Coefficient of }{\mathrm{x}}^5\text{ in }\left\{\frac{1}{\mathrm{x}}\left[(1+\mathrm{x}{)}^{31}-(1+\mathrm{x}{)}^{21}\right]\right\} \\ =\text{ Coefficient of }{\mathrm{x}}^6\text{ in }\left[(1+\mathrm{x}{)}^{31}-(1+\mathrm{x}{)}^{21}\right] \\ {=}^{31}{\mathrm{C}}_6{-}^{21}{\mathrm{C}}_6 \end{gathered}$$