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 

$\begin{array}{l}2S{=}^{23}{C}_0{+}^{23}{C}_1+{\dots }^{23}{C}_{23}=2^{23}\\ \Rightarrow S=2^{22}\end{array}$