The value of $$21C1$$−$$10C1+$$ $$21C2$$−$$10C2$$ $$+$$ $$21C3$$−$$10C3+$$ $$21C4$$−$$10C4+$$…$$++$$ $$21C10$$−$$10C10$$ is

Question

The value of  $$21C1$$−$$10C1+$$ $$21C2$$−$$10C2$$ $$+$$ $$21C3$$−$$10C3+$$ $$21C4$$−$$10C4+$$…$$++$$ $$21C10$$−$$10C10$$ is

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We have, $$21C1$$−$$10C1+$$ $$21C2$$−$$10C2+$$ $$21C3$$−$$10C3+$$ $$21C4$$−$$10C4+$$…$$+$$ $$21C10$$−$$10C10$$ $$21C1+21C2+21C3+21C4+$$…$$+21C10$$− $$10C1+10C2+10C3+10C4+$$…$$+10C10=12221C1+221C2+221C3+221C4+$$…$$+221C10$$− $$10C1+10C2+10C3+10C4+$$…$$+10C10=$$ $$21C1+21C2+21C3+21C4+$$…$$+21C10$$− $$10C1+10C2+10C3+10C4+$$…$$+10C10=12221C1+221C2+221C3+221C4+$$…$$+221C10=12$$ $$21C1+21C20+$$ $$21C2+21C19+$$ $$21C3+21C18+$$…$$+$$ $$21C10+21C11$$− $$10C1+10C2=12221$$− $$21C0+21C21$$−$$210$$−$$10C0=12221$$−$$2$$−$$210$$−$$1=220$$−$$210$$

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

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The value of 21C1−10C1+ 21C2−10C2 + 21C3−10C3+ 21C4−10C4+…++ 21C10−10C10 is

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