If $x^{n}$ occurs in the expansion of ${(x + 1 / x^{2})}^{2 n}$ , then the coefficient of $x^{n}$ is

\frac{(2 n!)}{(\frac{2 n - p}{3})! (\frac{4 n + p}{3})!}

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\frac{(2 n!)}{n! (2 n - p)!}

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\frac{2 n! 3! 3!}{(2 n - 1)!}

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None of these

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Solution

The correct option is D None of these
$(r + 1)$ th term in the expansion of $(x + 1 / x^{2})^{2 n}$ is given by,
$T_{r + 1} =^{2 n} C_{r} (x)^{r} (1 / x^{2})^{2 n - r} =^{2 n} C_{r} (x)^{r - 2 (2 n - r)} =^{2 n} C_{r} (x)^{3 r - 4 n}$
For coefficient of $x^{n}$ we have, $3 r - 4 n = n \Rightarrow r = \frac{5}{3} n$
Hence coefficient of $x^{n}$ is $^{2 n} C_{5 / 3 n} = \frac{(2 n)!}{(2 n - \frac{5}{3} n)! (\frac{5}{3} n)!} = \frac{(2 n)!}{(\frac{n}{3})! (\frac{5 n}{3})!}$
Also $r = 5 n / 3$ has to be integer $\Rightarrow n$ should be integral multiple of $3$

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