# True or False:The sum of series $\sum \limits_{r=0}^{10} 20C_r$ is $2^{19}+\large\frac{20C_{10}}{2}$

$\begin{array}{1 1}(A)\;True\\(B)\;False\end{array}$

Toolbox:
• $C(n,r)=\large\frac{n!}{r!(n-r)!}$
$\sum \limits_{r=0}^{10} 20C_r$
$20C_0+20C_1+20C_2+20C_3.............20C_{10}$
We have
$20C_0+20C_1+20C_2.......20C_{20}=2^{20}$
$\Rightarrow (20C_0+20C_1 +.......20C_{10})+(20C_{11}+.......20C_{20})=2^{20}$
$\Rightarrow (20C_0+......+20C_{10})+(20C_{9}+20C_8+......+20C_0)=2^{20}$
$\Rightarrow (20C_0+......+20C_{10})+(20C_{0}+......+20C_{10}-20C_{10})=2^{20}$
$\Rightarrow 2(20C_0+......+20C_{10})=2^{20}+20C_{10}$
$\Rightarrow 20C_0+......+20C_{10}=2^{19}+\large\frac{1}{2}$$20C_{10}$
Hence the given statement is True..