The coefficient of $x^{n}$ in the expansion of $(1 - 2 x + 3 x^{2} - 4 x^{3} + . . . .)^{- n}$ is?

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

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\frac{(2 n)!}{(n!)^{2}}

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\frac{1}{2} \frac{(2 n)!}{(n!)^{2}}

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None of these

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Solution

The correct option is B $\frac{(2 n)!}{(n!)^{2}}$
$(1 - 2 x + 3 x^{2} - 4 x^{3} + . . .)^{- n}$
$= ((1 + x)^{- 2})^{- n} = (1 + x)^{2 n}$
Coefficient of $x^{n}$ is $\frac{(2 n)!}{(n!) 2} =^{2 n} C_{n}$
$∴$ Coefficient of $x^{n}$ in $(1 - 2 x + 3 x^{2} . . .)^{- n} = \frac{(2 n)!}{(n!)^{2}}$

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