The smallest natural number n, such that the coefficient of $x$ in the expansion of ${(x^2+\frac{1}{x^3})}^n$ is ${ }^n{C}_{23},$ is

${(x^2+\frac{1}{x^3})}^n$

${\mathrm{T}}_{\mathrm{r}}=\sum _{\mathrm{r}=0}^{\mathrm{n}}{ }^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}{\mathrm{x}}^{2\mathrm{n}-2\mathrm{r}}\cdot {\mathrm{x}}^{-3\mathrm{r}}=\sum _{\mathrm{r}=0}^{\mathrm{n}}{ }^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}{\mathrm{x}}^{2\mathrm{n}-5\mathrm{r}}$

 has $T_{r+1}=n_{c_r}.x^{2n-5r}\Rightarrow 2n-5r=1$

$5r=2n-1;given{}^n{c}_r{=}^n{c}_{23}{=}^n{c}_{n-23}$

$r=23$ gives n = 58

R = n – 23 gives n = 38