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The sum of the series _{ r=0 }^{ 10 }{ _{ }^{ 20 }{ { C }_{

Question

The sum of the series $\sum_{r = 0}^{10}_{20} C_{r}$ is

A

2^{19} - \frac{1}{2} ._{20} C_{10}

B

2^{19} + \frac{1}{2} .^{20} C_{10}

C

2^{19}

D

2^{20}

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The sum of the series

\sum_{r = 0}^{10}_{20} C_{r}

is

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Text solutionVerified

We have,

\sum_{r = 0}^{10}^{20} C_{r} =^{20} C_{0} +^{20} C_{1} + ... +^{20} C_{10}

But

^{20} C_{0} +^{20} C_{1} + ... +^{20} C_{20} = 2^{20}

and

∵^{20} C_{20} =^{20} C_{0};^{20} C_{19} =^{20} C_{1};^{20} C_{18} =^{20} C_{2} ...

and

^{20} C_{11} =^{20} C_{9}

∴ r = 0 \sum 10^{20} C_{r} = (^{20} C_{0} +^{20} C_{1} + ... +^{20} C_{20}) - (^{20} C_{11} +^{20} C_{12} + ... +^{20} C_{20}) = 2^{20} +^{20} C_{10} - (^{20} C_{10} +^{20} C_{9} + ... +^{20} C_{0}) \Rightarrow 2 [^{20} C_{0} +^{20} C_{1} + ... +^{20} C_{10}] = 2^{20} +^{20} C_{10}

∴^{20} C_{0} +^{20} C_{1} + ... +^{20} C_{10} = 2^{19} + \frac{1}{2} .^{20} C_{10}

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Question Text	The sum of the series $\sum_{r = 0}^{10}_{20} C_{r}$ is
Answer Type	Text solution:1
Upvotes	150