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

$2^{21}-2^{11}$

$2^{21}-2^{10}$

$2^{20}-2^{9}$

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

Option D

$(^{21}C_{1}-^{10}C_{1})+(^{21}C_{2}-^{10}C_{2})+(^{21}C_{3}-^{10}C_{3})+.....+(^{21}C_{10}-^{10}C_{10})$

= $(^{21}C_{1}+^{21}C_{2}+....+^{21}C_{10})-(^{10}C_{1}+^{10}C_{2}+.......+^{10}C_{10})$

= $\frac{1}{2}(^{21}C_{1}+^{21}C_{2}+.....+^{21}C_{20})-(2^{10}-1)$

= $\frac{1}{2}(^{21}C_{1}+^{21}C_{2}+.....+^{21}C_{21}-1)-(2^{10}-1)$

= $\frac{1}{2}(2^{21}-2)=2^{20}-1-2^{10}+1-(2^{10}-1)$

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

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