( ( * 20 c 21 1 ) - ( * 20 c 10 1 ) ) + ( ( * 20 c 21 2 ) - ( * 2… - Vidyadip

MCQ

$\left( {\left( {\begin{array}{*{20}{c}}
{21}\\
1
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
1
\end{array}} \right)} \right) + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
2
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
2
\end{array}} \right)} \right)$$ + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
3
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
3
\end{array}} \right)} \right) + \;.\;.\;.$$ + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
{10}
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
{10}
\end{array}} \right)} \right) = $

✓
${2^{20}} - {2^{10}}$
B
${2^{21}} - {2^{11}}$
C
${2^{21}} - {2^{10}}$
D
${2^{20}} - {2^9}$

✓

Answer

Correct option: A.

${2^{20}} - {2^{10}}$

a
We have $\left(^{21} \mathrm{C}_{1}+^{21} \mathrm{C}_{2} \ldots \ldots+^{21} \mathrm{C}_{10}\right)$

$-\left(^{10} \mathrm{C}_{1}+^{10} \mathrm{C}_{2} \ldots . .^{10} \mathrm{C}_{10}\right)$

$=\frac{1}{2}\left[\left(^{21} \mathrm{C}_{1}+\ldots+^{21} \mathrm{C}_{10}\right)+\left(^{21} \mathrm{C}_{11}+\ldots^{21} \mathrm{C}_{20}\right)\right]-\left(2^{10}-1\right)$

$\left(\because 10 \mathrm{C}_{1}+^{10} \mathrm{C}_{2}+\ldots .+^{10} \mathrm{C}_{10}=2^{10}-1\right)$

$=\frac{1}{2}\left[2^{21}-2\right]-\left(2^{10}-1\right)$

$=\left(2^{20}-1\right)-\left(2^{10}-1\right)=2^{20}-2^{10}$

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