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Expansions of the Form (1+x)^(-n) and (1-x)^(-n)

In the expans...

Question

In the expansion of $(1 + x + x^{2} + x^{3} + x^{4})^{- 2}$ , coefficient of $x^{10}$ is

3

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Solution

The correct option is A $3$
$(1 + x + x^{2} + x^{3} + x^{4})^{- 2} = (\frac{1 (1 - (x)^{5})}{1 - x})^{- 2} = (1 - x)^{2} (1 - x^{5})^{- 2}$
$= (1 - 2 x + x^{2}) (1 - (- 2) x^{5} + \frac{(- 2) (- 2 - 1)}{2} x^{10} + \dots)$

Co efficient of $x^{10}$ will be $\frac{(- 2) (- 3)}{2} \times 1 = 3$

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