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  4. The sum of the series 20C0 - 20C1 + 20C2 - 20C3 + ..... - .... + 20C10 is

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Q. The sum of the series ${^{20}C_0} - {^{20}C_1} + {^{20}C_2} - {^{20}C_3} + ..... - .... + {^{20}C_{10}}$ is

AIEEEAIEEE 2007Binomial Theorem

Solution:

$\left(1+x\right)^{20}=^{20}C_{0}+^{20}C_{1}x+...+^{20}C_{10}x^{10}+...+^{20}C_{20}x^{20}$
put $x = − 1,$
$0=^{20}C_{0}-^{20}C_{1}+...-^{20}C_{9}+^{20}C_{11}+...+^{20}C_{20}$
$0=2\left(^{20}C_{0}-^{20}C_{1}+...-^{20}C_{9}\right)=^{20}C_{10}$
$\Rightarrow ^{20}C_{0}-^{20}C_{1}+...+^{20}C_{10}=\frac{1}{2}^{20}C_{10}.$