Q59 / 160 · TS EAMCET 2024 (10 May Shift 2) · Mathematics

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
    $(25) \mathrm{S}$
  • Correct answer
    C
    $(150) \mathrm{S}$
  • D
    $(100) \mathrm{S}$

Solution

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. $x$ $100\left(1+x+x^2+x^3\right)^{99}\left(1+2 x+3 x^2\right)=\Sigma \mathbb{I} a_r$ $\begin{aligned} & \text { Put } x=1 ; \Sigma \mathbb{I} \mathrm{a}_{\mathrm{r}}=100\left(4^{99}\right)(6)=(600)(4)^{39}=\frac{600}{4}(4)^{1000} \\ & \Sigma r a_r=150 \mathrm{~S}\end{aligned}$

Hence, the correct answer is option: 'C'.

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