The value of the term independent of x in the expansion of ${(\begin{matrix} x^{2} - \frac{1}{x} \end{matrix})}^{27}$ is :

9

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18

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48

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84

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Solution

The correct option is A $9$
General term of a expansion ${(a + b)}^{n}$ is $t_{r + 1} =^{n} C_{r} . a^{r} . b^{n - r}$
So $t_{r + 1} =^{n} C_{r} . {(x^{2})}^{r} . {(\frac{- 1}{x})}^{n - r} \Rightarrow t_{r + 1} =^{n} C_{r} . x^{2 r - (n - r)}$
Term independent of $x$ will be when power of $x$ is zero and that is
$2 r - n + r = 0 \Rightarrow n = 3 r r = 27 / 3 = 9$

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