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  • The coefficient of  in the expression <img alt="(x-1)\left(x^2-2\right)\left(x^3-3\right) \ldots \

The coefficient of x^{203} in the expression (x-1)\left(x^2-2\right)\left(x^3-3\right) \ldots \ldots .\left(x^{20}-20\right) must be:

Option: 1

11


Option: 2

12


Option: 3

13


Option: 4

15


Answers (1)

best_answer

Expression

$=$ $$ x \cdot x^2 \cdot x^3 \ldots \ldots x^{20}\left(1-\frac{1}{x}\right)\left(1-\frac{2}{x^2}\right)\left(1-\frac{3}{x^3}\right) \cdots\left(1-\frac{20}{x^{20}}\right) $$ $$ =x^{210} \cdot E

where E=\left(1-\frac{1}{x}\right)\left(1-\frac{2}{x^2}\right)\left(1-\frac{3}{x^3}\right) \ldots . .\left(1-\frac{20}{x^{20}}\right)
Now, coefficient of x^{203}  in original expression = coefficient of x^{-7} in E
But,

\begin{aligned} E & =1-\left(\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}+\ldots \ldots\right)+\left(\frac{1}{x} \cdot \frac{6}{x^6}+\frac{2}{x^2} \cdot \frac{5}{x^5}+\frac{3}{x^3} \cdot \frac{4}{x^4}+\ldots \ldots\right)-\left(\frac{1}{x} \cdot \frac{2}{x^2} \cdot \frac{4}{x^4}+\ldots \ldots .\right) \\ & =\text { coefficient of } x^{-7}=-7+6+10+12-8=13 \end{aligned}

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Divya Prakash Singh

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