JEE Mains · Maths · STD 11 - 7. binomial theoram
The coefficient of \(x ^{301}\) in \((1+x)^{500}+x(1+x)^{499}+x^2(1+x)^{498}+\ldots . .+x^{500}\) is:
- A \({ }^{501} C _{302}\)
- B \({ }^{500} C _{301}\)
- C \({ }^{500} C _{300}\)
- D \({ }^{501} C _{200}\)
Answer & Solution
Correct Answer
(D) \({ }^{501} C _{200}\)
Step-by-step Solution
Detailed explanation
\((1+x)^{500}+x(1+x)^{499}+x^2(1+x)^{498}+\ldots .+x^{500}\) \(=(1+x)^{500} \cdot\left\{\frac{1-\left(\frac{x}{1+x}\right)^{501}}{1-\frac{x}{1+x}}\right\}\) \(=(1+x)^{500} \frac{\left((1+x)^{501}-x^{501}\right)}{(1+x)^{501}} \cdot(1+x)\) \(=(1+x)^{501}-x^{501}\) Coefficient of…
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