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

Let

$\begin{array}{l} S=(1+x)^{21}+(1+x)^{22}+(1+x)^{23}+\ldots(1+x)^{30} \ldots(\mathrm{i}) \\ (1+x) S=(1+x)^{22}+(1+x)^{23} \ldots(1+x)^{30}+(1+x)^{31} \end{array}$

Subtracting (i) from (ii), we get

$x S=(1+x)^{31}-(1+x)^{21}$

$S=\frac{(1+x)^{31}}{x}-\frac{(1+x)^{21}}{x}$

Coefficient of $x^{r}$

$\begin{array}{l} ={ }^{31} C_{r} x^{r-1}-{ }^{21} C_{r} x^{r-1} \\ =\left({ }^{31} C_{r}-{ }^{21} C_{r}\right) x^{r-1} \ldots(a) \end{array}$

Therefore the coefficient of $x^{5}$ implies

$r-1=5$

$r=6$

Substituting in (a), we get a coefficient of $x^{5}$

$=\left({ }^{31} C_{6}-{ }^{21} C_{6}\right)$