Question:
The sum of the series ${^{20}C_0} - {^{20}C_1} + {^{20}C_2} - {^{20}C_3} + ..... - .... + {^{20}C_{10}}$ is
The sum of the series ${^{20}C_0} - {^{20}C_1} + {^{20}C_2} - {^{20}C_3} + ..... - .... + {^{20}C_{10}}$ is
Updated On: Jul 5, 2022
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- ${^{20}C_{10}}$
- ${^{ - 20}C_{10}}$
- $ \frac{1}{2} {^{20}C_{10}}$
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The Correct Option is D
Solution and Explanation
$\left(1+x\right)^{20}=^{20}C_{0}+^{20}C_{1}x+...+^{20}C_{10}x^{10}+...+^{20}C_{20}x^{20}$
put $x = - 1,$
$0=^{20}C_{0}-^{20}C_{1}+...-^{20}C_{9}+^{20}C_{11}+...+^{20}C_{20}$
$0=2\left(^{20}C_{0}-^{20}C_{1}+...-^{20}C_{9}\right)=^{20}C_{10}$
$\Rightarrow ^{20}C_{0}-^{20}C_{1}+...+^{20}C_{10}=\frac{1}{2}^{20}C_{10}.$
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
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- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.