TS EAMCET · Maths · Binomial Theorem
If the coefficient of \(x^r\) in the expansion of \(\left(1+x+x^2+x^3\right)^{100}\) is \(a_r\), and \(S=\sum_{r=0}^{300} a_r\) then \(\sum_{r=0}^{300} r \cdot a_r=\)
- A \((50) \mathrm{S}\)
- B \((25) \mathrm{S}\)
- C \((150) \mathrm{S}\)
- D \((100) \mathrm{S}\)
Answer & Solution
Correct Answer
(C) \((150) \mathrm{S}\)
Step-by-step Solution
Detailed explanation
Put \(x=1\) in \(\left(1+x+x^2+x^3\right)^{100}\) Then \(S=4^{100}\) Differentiating \(\left(1+x+x^2+x^3\right)^{100}\) w.x.t. \(x\) \(100\left(1+x+x^2+x^3\right)^{99}\left(1+2 x+3 x^2\right)=\Sigma \mathbb{I} a_r\)…
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