The sum of the series ( * 20 c 20 0 ) - ( * 20 c 20 1 )+ ( * 20 c… - Vidyadip

MCQ

The sum of the series $\left( {\begin{array}{*{20}{c}}{20}\\0\end{array}} \right) - \left( {\begin{array}{*{20}{c}}{20}\\1\end{array}} \right)$$+$$\left( {\begin{array}{*{20}{c}}{20}\\2\end{array}} \right) - \left( {\begin{array}{*{20}{c}}{20}\\3\end{array}} \right)$$+…..-……+$$\left( {\begin{array}{*{20}{c}}{20}\\{10}\end{array}} \right)$

A
$0$
B
$\;\left( {\begin{array}{*{20}{c}}{20}\\{10}\end{array}} \right)$
C
-$\;\left( {\begin{array}{*{20}{c}}{20}\\{10}\end{array}} \right)$
✓
$\frac{1}{2}\left( {\begin{array}{*{20}{c}}{20}\\{10}\end{array}} \right)$

✓

Answer

Correct option: D.

$\frac{1}{2}\left( {\begin{array}{*{20}{c}}{20}\\{10}\end{array}} \right)$

d
We know that, $(1+x)^{20}=20 \mathrm{C}_{0}+20 \mathrm{C}_{1} \mathrm{x}+^{20} \mathrm{C}_{2}$

$\mathrm{x}^{2}+\ldots \ldots .20 \mathrm{C}_{10} \mathrm{c}_{10} \mathrm{x}^{10}+\ldots . .2 .20 \mathrm{C}_{20} \mathrm{x}^{20}$

Put $\mathrm{x}=-1 .(0)= ^{20} \mathrm{C}_{0}- ^{20} \mathrm{C}_{1}+ ^{20} \mathrm{C}_{2}- ^{20} \mathrm{C}_{3}+$$\ldots . .+ ^{20} \mathrm{C}_{10}- ^{20} \mathrm{C}_{11} \ldots+^{20} \mathrm{C}_{20}$

$0=2[^{20} \mathrm{C}_{0}- ^{20} \mathrm{C}_{1}+ ^{20} \mathrm{C}_{2}- ^{20} $$\mathrm{C}_{3}+\ldots . .- ^{20} \mathrm{C}_{9} ]+ ^{20} \mathrm{C}_{10}$

$^{20} \mathrm{C}_{10}=2 [^{20} \mathrm{C}_{0}-^{20} \mathrm{C}_{1}$$+ ^{20} \mathrm{C}_{2}-^{20} \mathrm{C}_{3}+\ldots \ldots-^{20} \mathrm{C}_{9}+^{20} \mathrm{C}_{10}]$

$^{20} \mathrm{C}_{0}-^{20} \mathrm{C}_{1}+^{20} \mathrm{C}_{2}-^{20} \mathrm{C}_{3} \cdot \ldots .$ i" $^{2 \mathrm{n}} \mathrm{C}_{10}$

$=\frac{1}{2}^{20} \mathrm{C}_{10}$

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