Question

The value of the expression $\left(\displaystyle\sum_{r=0}^{10}{ }^{10} C _r\right)\left(\displaystyle\sum_{K=0}^{10}(-1)^K \frac{{ }^{10} C _{ K }}{2^{ K }}\right)$ is :

• A. $2^{10}$
• B. $2^{20}$
• C. 1
• D. $2^5$

Answer 1

Answer: (C)

$\left(\displaystyle\sum_{r=0}^{10}{ }^{10} C_r\right)\left(\displaystyle\sum_{k=0}^{10}(-1)^k \frac{{ }^{10} C_k}{2^k}\right)$
$=\left({ }^{10} C _0+\ldots \ldots . .+{ }^{10} C _{10}\right)\left({ }^{10} C _0-\frac{{ }^{10} C _1}{2}+\frac{{ }^{10} C _2}{2^2} \ldots \ldots . .+\frac{{ }^{10} C _{10}}{2^{10}}\right)$
$=2^{10} \times\left(1-\frac{1}{2}\right)^{10}=1$