The natural number m , for which the coefficient of x in the binomial expansion of x m + 1 x 2 22 is 1540, is

The general term in the binomial expansion,

${\mathrm{T}}_{\mathrm{r}+1}={}^{22}C_r{\left({\mathrm{x}}^{\mathrm{m}}\right)}^{22-\mathrm{r}}{x}^{-2r}$

$\Rightarrow {\mathrm{T}}_{\mathrm{r}+1}={}^{22}\mathrm{C}_{\mathrm{r}}{\mathrm{x}}^{\mathrm{m}(22-r)-2\mathrm{r}}$

For the coefficient of $x$

$22 m-m r-2 r=1$

$\Rightarrow 22 m-1=r(m+2)$

$\Rightarrow r=\frac{22m-1}{m+2}$

$\Rightarrow r=\frac{22m+44-45}{m+2}$

$r=22-\frac{3.3.5}{m+2}\Rightarrow 22-r=\frac{3·3·5}{m+2}$

Since, ${}^{22}C_r={}^{22}C_{22-r}$

So possible values of $\mathrm{m}=1,3,7,13,43$, but ${}^{22}\mathrm{C}_3=1540\Rightarrow 22-r=3\Rightarrow m=13$