TS EAMCET · Maths · Binomial Theorem
If \(n\) is a positive integer, then the coefficient of \(x^6\) in the expansion of \(\left(1-2 x+3 x^2-4 x^3+\ldots\right)^{-n}\) is
- A \({ }^{(2 n)} \mathrm{C}_4\)
- B \({ }^n \mathrm{C}_{12}\)
- C \({ }^{(2 n)} \mathrm{C}_6\)
- D \({ }^n C_6\)
Answer & Solution
Correct Answer
(C) \({ }^{(2 n)} \mathrm{C}_6\)
Step-by-step Solution
Detailed explanation
We know that, \[ \begin{aligned} &(1+x)^{-2}=1-2 x+3 x^2-4 x^3+\ldots \\ & \therefore \quad\left(1-2 x+3 x^2-4 x^3+\ldots\right)^{-n} \\ &=\left[(1+x)^{-2}\right]^{-n}=(1+x)^{2 n} . \end{aligned} \] Now, the general term in the expansion of \((1+x)^{2 n t}\) is…
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