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

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

$S =^{23} C_{11} +^{23} C_{10} + \dots +^{23} C_{0}$

Adding (1) and (2), we get

$\begin{array}{l} 2 S =^{23} C_{0} +^{23} C_{1} + \dots^{23} C_{23} = 2^{23} \\ \Rightarrow S = 2^{22} \end{array}$

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