Question

The value of $\sum _{r=0}^{10}\left(r\right)$ ${}^{20}{C}_r$ is equal to?

• A. $20(2^{18}+{}^{19}{C}_{10})$
• B. $10(2^{18}+{}^{19}{C}_{10})$
• C. $20(2^{18}+{}^{19}{C}_{11})$
• D. $10(2^{18}+{}^{19}{C}_{11})$

Answer 1

The correct option is A $20(2^{18}+{}^{19}{C}_{10})$

$\begin{array}{c}{\sum }_{r=0}^{10}\left(r\right){}^{20}{C}_r\\ \Rightarrow 0{+}^{20}{C}_1+2^{20}{C}_2+3^{20}{C}_3+............+{10}^{20}{C}_{10}\\ \Rightarrow 20+20\times 19+\frac{20\times 19\times 18}{2}+.........+10\times \frac{20!}{10!\times 10!}\\ \Rightarrow 20\left[1+19+\frac{19\times 18}{2}+\frac{19\times 18\times 17}{6}+...........+\frac{10\times 19!}{10!\times 10!}\right]\\ \Rightarrow 20\left[1+19+\frac{19\times 18}{2}+\frac{19\times 18\times 17}{6}+.........+\frac{19!}{9!\times 10!}\right]\\ 20\left[{}^{19}{C}_0{+}^{19}{C}_1{+}^{19}{C}_2{+}^{19}{C}_3+..........{+}^{19}{C}_{10}\right]Ans.\end{array}$