Question

The value of $\sum _{k=0}^{10}{}^{20}{C}_k$ is equal to

• A. $2^{19}+\frac{1}{2}$ ${}^{20}{C}_{10}$
• B. $2^{19}$
• C. ${}^{20}{C}_{10}$
• D. None of these

Answer 1

The correct option is A $2^{19}+\frac{1}{2}$ ${}^{20}{C}_{10}$

The value of $\sum _{k=0}^{10}{}^{20}{C}_k{=}^{20}{C}_0{+}^{20}{C}_1+\cdots {+}^{20}{C}_{10}$ is
$=\frac{1}{2}[2{\cdot }^{20}{C}_0+2{\cdot }^{20}{C}_1+\cdots +2{\cdot }^{20}{C}_{10}]$
$=\frac{1}{2}[\left({}^{20}{C}_0{+}^{20}{C}_{20}\right)+\left({}^{20}{C}_1{+}^{20}{C}_{19}\right)+\cdots +2{\cdot }^{20}{C}_{10}]$
$=\frac{1}{2}\left[({}^{20}{C}_0{+}^{20}{C}_1{+}^{20}{C}_2+\cdots {+}^{20}{C}_{19}{+}^{20}{C}_{20}){+}^{20}{C}_{10}\right]$
$=\frac{1}{2}\left[2^{29}{+}^{20}{C}_{10}\right]$
$=2^{19}+{\frac{1}{2}}^{20}{C}_{10}$