Mathematics › Binomial Theorem › Sum of Series
The sum of the series ${ }^{20} \mathrm{C}_0-{ }^{20} \mathrm{C}_1+{ }^{20} \mathrm{C}_2-{ }^{20}…
The sum of the series
${ }^{20} \mathrm{C}_0-{ }^{20} \mathrm{C}_1+{ }^{20} \mathrm{C}_2-{ }^{20} \mathrm{C}_3+\ldots-\ldots+{ }^{20} \mathrm{C}_{10} \text { is }$
$-{ }^{20} \mathrm{C}_{10}$
$\frac{1}{2}{ }^{20} \mathrm{C}_{10}$
$0$
${ }^{20} \mathrm{C}_{10}$ Solution
$(1+\mathrm{x})^{20}={ }^{20} \mathrm{C}_0+{ }^{20} \mathrm{C}_1 \mathrm{x}+\ldots+{ }^{20} \mathrm{C}_{10} \mathrm{x}{ }^{10}+\ldots+{ }^{20} \mathrm{C}_{20} \mathrm{x}^{20}$
put $\mathrm{x}=-1$,
$0={ }^{20} \mathrm{C}_0-{ }^{20} \mathrm{C}_1+\ldots-{ }^{20} \mathrm{C}_9+{ }^{20} \mathrm{C}_{10}-{ }^{20} \mathrm{C}_{11}+\ldots+{ }^{20} \mathrm{C}_{20}$
$0=2\left({ }^{20} \mathrm{C}_0-{ }^{20} \mathrm{C}_1+\ldots-{ }^{20} \mathrm{C}_9\right)+{ }^{20} \mathrm{C}_{10}$
$\Rightarrow{ }^{20} \mathrm{C}_0-{ }^{20} \mathrm{C}_1+\ldots+{ }^{20} \mathrm{C}_{10}=\frac{1}{2}{ }^{20} \mathrm{C}_{10}$.
Asked in: JEE Main 2007
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