The coefficient of $x^{50}$ in the expansion of $(1+x)^{100}+2 x(1+x)^{99}+3 x^2(1+x)^{98}+$ $+101 x^{100}$,…

The coefficient of $x^{50}$ in the expansion of $(1+x)^{100}+2 x(1+x)^{99}+3 x^2(1+x)^{98}+$ $+101 x^{100}$, is
  1. ${ }^{100} \mathrm{C}_{50}$
  2. ${ }^{101} \mathrm{C}_{50}$
  3. ${ }^{102} \mathrm{C}_{50}$
  4. ${ }^{103} \mathrm{C}_{50}$

Solution

$ \begin{aligned} & \text { Let } S=(1+x)^{100}+2 x(1+x)^{99}+3 x^2(1+x)^{98} \\ & +\ldots+101 x^{100} \\ & \frac{x}{1+x} S= \\ & x(1+x)^{99}+2 x^2(1+x)^{98}+\ldots+100 x^{100}+101 \frac{x^{101}}{1+x} \\ & \Rightarrow \frac{S}{1+x}=(1+x)^{100}+x(1+x)^{99}+x^2(1+x)^{98} \\ & +\ldots+x^{100}-101 \frac{x^{101}}{1+x} \\ & =\frac{(1+x)^{100}\left[\left(\frac{x}{1+x}\right)^{101}-1\right]}{\frac{x}{1+x}-1}-101 \frac{x^{101}}{1+x} \\ & \Rightarrow \quad S=(1+x)^{102}-x^{102}-102 x^{101} \\ & \end{aligned} $ So, coefficient of $x^{50}$ in the expansion of $S={ }^{102} C_{50}$

Asked in: AP EAMCET 2018 (23 Apr Shift 2)

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