JEE Mains · Maths · STD 11 - 7. binomial theoram
The coefficient of \(x ^{101}\) in the expression \((5+x)^{500}+x(5+x)^{499}+x^{2}(5+x)^{498}+\ldots . x^{500}\) \(x>0\), is
- A \({ }^{501} C _{101}(5)^{399}\)
- B \({ }^{501} C _{101}(5)^{400}\)
- C \({ }^{501} C _{100}(5)^{400}\)
- D \({ }^{500} C _{101}(5)^{399}\)
Answer & Solution
Correct Answer
(A) \({ }^{501} C _{101}(5)^{399}\)
Step-by-step Solution
Detailed explanation
\((5+x)^{500}+x(5+x)^{499}+x^{2}(5+x)^{498}+\ldots+x^{500}\) \(=\frac{(5+x)^{501}-x^{501}}{(5+x)-x}=\frac{(5+x)^{501}-x^{501}}{5}\) \(\Rightarrow\) coefficient \(x ^{101}\) in given expression \(=\frac{{ }^{501} C _{101} 5^{400}}{5}={ }^{501} C _{101} 5^{399}\)
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