Question

The value of
$(^{21}C_{1} - ^{10}C_{1}) + (^{21}C_{2} - ^{10}C_{2}) + (^{21}C_{3} - ^{10}C_{3}) +(^{21}C_{4} - ^{10}C_{4}) +....+(^{21}C_{10} - ^{10}C_{10})$ is :

• A. $2^{21} - 2^{10}$
• B. $2^{20} - 2^{9}$
• C. $2^{20} - 2^{10}$
• D. $2^{21} - 2^{11}$

Answer 1

Answer: (C)

$= ^{21}C_{1}+^{21}C_{2}+...+^{21}C_{10} \frac{1}{2}\left\{^{21}C_{0}+^{21}C_{1}+...+^{21}C_{21}\right\}-1$
$= 2^{20}-1$
$\left(^{10}C_{1}+^{10}C_{2}+...+^{10}C_{10}\right) = 2^{10}-1$
$\therefore $ Required sum $= \left(2^{20} - 1\right) - \left(2^{10} - 1\right)$
$= 2^{20} - 2^{10}$