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Find the sum of^{21}C_{0} +^{21}C_{1} +^{21}C_{2} +^{21}C_{3} +^{21}C_{4} + .......... +^{21}C_{10}.

Please find below the solution to the asked query:

${}^{21}{C}_{0}{+}^{21}{C}_{1}{+}^{21}{C}_{2}{+}^{21}{C}_{3}{+}^{21}{C}_{4}+...{+}^{21}{C}_{10}\phantom{\rule{0ex}{0ex}}=1+21+\frac{21\times 20}{1\times 2}+\frac{21\times 20\times 19}{1\times 2\times 3}+\frac{21\times 20\times 19\times 18}{1\times 2\times 3\times 4}+...+\frac{21\times 20\times 19\times 18\times ...\times 12}{1\times 2\times 3\times 4\times ...\times 10}$

Hope this information will clear your doubts about Binomial theorem.

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