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

${\left(1+x\right)}^{21}+{\left(1+x\right)}^{22}+\dots \dots +{\left(1+x\right)}^{30}$

$={\left(1+x\right)}^{21}\left[1+{\left(1+x\right)}^1+\dots +{\left(1+x\right)}^9\right]$

$={\left(1+x\right)}^{21}\left[\frac{{\left(1+x\right)}^{10}-1}{\left(1+x\right)-1}\right]$

$=\frac{1}{x}\left[{\left(1+x\right)}^{31}-{\left(1+x\right)}^{21}\right]$

$\therefore$ Coefficient of $x^5$ in the given expression

$=$ Coefficient of $x^5$ in $\frac{1}{x}\left[{\left(1+x\right)}^{31}-{\left(1+x\right)}^{21}\right]$

$=$ Coefficient of $x^6$ in $\left[{\left(1+x\right)}^{31}-{\left(1+x\right)}^{21}\right]$

$={}^{31}C_6-{}^{21}C_6$