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Q.
The coefficient of $x^{50}$ in the expansion of $\left(1+x\right)^{100} + 2x \left(1+x\right)^{99} + 3x^{2} \left(1+x\right)^{98} + ... + 101x^{100}, $ is
Let $S=(1+x)^{100}+2 x(1+x)^{99}+3 x^{2}(1+x)^{98}$
$+\ldots+101 x^{100}$
$\frac{x}{1+x} S=$
$x(1+x)^{99}+2 x^{2}(1+x)^{98}+\ldots+100 x^{100}+101 \frac{x^{101}}{1+x}$
$\Rightarrow \frac{S}{1+x}=(1+x)^{100}+x(1+x)^{99}+x^{2}(1+x)^{98}$
$+\ldots+x^{100}-101 \frac{x^{101}}{1+x}$
$\Rightarrow S=(1+x)^{102}-x^{102}-102 x^{101}$
So, coefficient of $x^{50}$ in the expansion of $S={ }^{102} C_{50}$