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The sum of the series ^{20}C_0 - ,^{20}C_1+,^{20}C_2-,^{20}C_3

Question

The sum of the series $^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + ... +^{20} C_{10}$ is

-^{20} C_{10}

\frac{1}{2}^{20} C_{10}

0

^{20} C_{10}

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The sum of the series

^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + ... +^{20} C_{10}

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Text solutionVerified

On putting

x = - 1

(1 + x)^{20} =^{20} C_{0} +^{20} C_{1} x + ... + 20 C_{10} x^{10} + ... +^{20} C_{20} x^{20}

We get,

0 =^{20} C_{0} -^{20} C_{1} + ... -^{20} C_{9} +^{20} C_{10} -^{20} C_{11} + ... +^{20} C_{20}

\Rightarrow 0 =^{20} C_{0} -^{20} C_{1} + ... -^{20} C_{9} +^{20} C_{10} -^{20} C_{9} + ... +^{20} C_{0}

........

[∵^{n} C_{r} =^{n} C_{n - r}]

\Rightarrow 0 = 2 (^{20} C_{0} -^{20} C_{1} + ... -^{20} C_{9}) +^{20} C_{10}

\Rightarrow^{20} C_{10} = 2 (^{20} C_{0} -^{20} C_{1} + ... +^{20} C_{10})

.... [Adding

^{20} C_{10}

on both the sides]

\Rightarrow^{20} C_{0} -^{20} C_{1} + ... +^{20} C_{10} = \frac{1}{2}^{20} C_{10}

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