Binomial theorem for positive integral Index

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Question

Sum of the last $12$ coefficients of in the binomial expansion of ${(1 + x)}^{23}$ is:

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Solution

Sum of last $12$ coefficients is
$S =^{23} C_{12} +^{23} C_{13} + \dots +^{23} C_{23}$
Using $^{n} C_{r} =^{n} C_{n - r}$ we get
$\begin{array}{l} 2 S =^{23} C_{0} +^{23} C_{1} + \dots^{23} C_{23} = 2^{23} \\ \Rightarrow S = 2^{22} \end{array}$

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