The sum of the series $\sum _{r=0}^{10}{ }^{20}{\mathrm{C}}_{r\mathrm{\prime }}$ is

$\sum _{r=0}^{10}{ }^{20}{C}_r$

$\begin{array}{l}\left({ }^{20}{C}_0{+}^{20}{C}_1+\dots {+}^{20}{C}_{10}\right)\\ =\frac{1}{2}\left\{2\left({ }^{20}{C}_0{+}^{20}{C}_1+\dots {+}^{20}{C}_{10}\right)\right\}\\ =\frac{2}{2}\left\{\left({ }^{20}{C}_0{+}^{20}{C}_1+\dots {+}^{20}{C}_{10}\right)+\left({ }^{20}{C}_{20}{+}^{20}{C}_{19}+\dots {+}^{20}{C}_{10}\right)\right\}\\ =\frac{1}{2}\left\{\left({ }^{20}{C}_0{+}^{20}{C}_1+\dots {+}^{20}{C}_{10}{+}^{20}{C}_{11}+\dots {+}^{20}{C}_{20}\right){+}^{20}{C}_{10}\right\}\end{array}$

$=\frac{1}{2}\left\{2^{20}{+}^{20}{C}_{10}\right\}=2^{19}+{\frac{1}{2}}^{20}{C}_{10}$