The coefficient of in the expansion of , , is: [2026]

Question

The coefficient of $x^{2}$ in the expansion of ${(2 x^{2} + \frac{1}{x})}^{10}$ , $x \neq 0$ , is: [2026]

Smart Achievers · Accepted Answer

(2)

We have, ${(2 x^{2} + \frac{1}{x})}^{10}, x \neq 0$

$T_{r + 1} = {}^{10}C_{r} {(2 x^{2})}^{10 - r} {(\frac{1}{x})}^{r}$

$\Rightarrow T_{r + 1} = {}^{10}C_{r} {(2)}^{10 - r} {\cdot (x)}^{20 - 3 r}$

We have to find coefficient of $x^{2}$

$∴ 20 - 3 r = 2 \Rightarrow 3 r = 18 \Rightarrow r = 6$

$∴ Coefficient of x^{2} = {}^{10}C_{6} {(2)}^{10 - 6}$

$\Rightarrow {}^{10}C_{6} {(2)}^{4} = 210 \times 16 = 3360$

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