Question

The sum $\frac{1}{2}$ ${}^{10}{\text{C}}_0-$ ${}^{10}{\text{C}}_1+$ $2{\cdot }^{10}{\text{C}}_2-$ $2^2{\cdot }^{10}{\text{C}}_3$+...+$2^9{\cdot }^{10}{\text{C}}_{10}$ is equal to

• A. $\frac{1}{2}$
• B. $0$
• C. $\frac{1}{2}\cdot 3^{10}$
• D. none of these

Answer 1

The correct option is A $\frac{1}{2}$

Let
$S_n=\frac{1}{2}{}^{10}{C}_0-{}^{10}{C}_1+...{}^{10}{C}_{10}{2}^9$
$2S_n={}^{10}{C}_0-2{}^{10}{C}_1+2^2{}^{10}{C}_2+...{}^{10}{C}_{10}{2}^{10}$
$2S_n=(1-2{)}^{10}$
$2S_n=\left(1\right)$
$S_n=\frac{1}{2}$