The natural number m , for which the coefficient of x in the...

The general term in the binomial expansion,

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

$⇒ {T}_{r + 1} = Cr 22 {x}^{m ( 22 - r ) - 2r}$

For the coefficient of $x$

$22m - mr - 2r = 1$

$⇒ 22m - 1 = r ( m + 2 )$

$⇒ r = \dfrac{22m - 1}{m + 2}$

$⇒ r = \dfrac{22m + 44 - 45}{m + 2}$

$r = 22 - \dfrac{3 . 3 . 5}{m + 2} ⇒ 22 - r = \dfrac{3 \cdot 3 \cdot 5}{m + 2}$

Since, $Cr 22 = C22 - r 22$

So possible values of $m = 1, 3, 7, 13, 43$, but $C3 22 = 1540 ⇒ 22 - r = 3 ⇒ m = 13$