$\text{ Sum of the last }20\text{ coefficients in the expansion of }(1+x{)}^{39}\text{ , when expanded }$$\text{ in ascending powers of }\mathrm{x},\text{ is }$

$\text{ There are }40\text{ terms in the expansion of }(1+x{)}^{39}\text{ :Sum of last }20$ $\text{coefficients is }$

$\begin{array}{l}S{=}^{39}{\mathrm{C}}_{20}{+}^{39}{\mathrm{C}}_{21}+\dots {+}^{39}{\mathrm{C}}_{38}{+}^{39}{\mathrm{C}}_{39}\\ \Rightarrow S{=}^{39}{\mathrm{C}}_{19}{+}^{39}{\mathrm{C}}_{18}+\dots {+}^{39}{\mathrm{C}}_1{+}^{39}{\mathrm{C}}_0\\ \text{ Adding the above two expansions we get }2S{=}^{39}{\mathrm{C}}_0{+}^{39}{\mathrm{C}}_1+\dots {+}^{39}{\mathrm{C}}_{39}\\ =2^{39}\Rightarrow S=2^{38}.\end{array}$