Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • The coefficient of  in the expression <img alt="(x-1)\left(x^2-2\right)\left(x^3-3\right) \ld

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

\begin{aligned} \text { 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) \ldots \ldots \cdot\left(1-\frac{20}{x^{20}}\right) \\ & =x^{210} \cdot E \end{aligned}

\text { 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,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

Posted by

himanshu.meshram

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Latest Question