Question:
The natural number $m$, for which the coefficient of $x$ in the binomial expansion of $\left(x^{m}+\frac{1}{x^{2}}\right)^{22}$ is 1540 , is__________.
Solution:
$T_{r+1}={ }^{22} C_{r} \cdot\left(x^{m}\right)^{22-r} \cdot\left(\frac{1}{x^{2}}\right)^{r}$
$T_{r+1}={ }^{22} C_{r} \cdot x^{22 m-m r-2 r}$
$\because 22 m-m r-2 r=1$
$\Rightarrow r=\frac{22 m-1}{m+2} \Rightarrow r=22-\frac{3 \cdot 3 \cdot 5}{m+2}$
So, possible value of $m=1,3,7,13,43$
But ${ }^{22} C_{r}=1540$
$\therefore$ Only possible value of $m=13$
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