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  4. The smallest natural number n, such that the coefficient of x in the expansion of (x2 + (1/x3))n is nC23 , is :

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Q. The smallest natural number n, such that the coefficient of $x$ in the expansion of $\left(x^{2} + \frac{1}{x^{3}}\right)^{n} $ is $^nC_{23}$ , is :

JEE MainJEE Main 2019Binomial Theorem

Solution:

$T_{r} = \sum^{n}_{r=0} {^{n}C_{r} } x^{2n-2r} .x^{-3r} $
2n - 5r = 1 $\Rightarrow $ $2n = 5r + 1 $
for $r = 15$ $n = 38$ smallest value of $n$ is $38$.