The sum of the series $^{20} C_{0} -^{20} C_{1} -^{20} C_{2} -^{20} C_{3} . . . . . . . .^{20} C_{10} i s$

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(\frac{1}{2})^{20} C_{10}

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^{- 20} C_{10}

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(\frac{1}{3})^{20} C_{10}

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Solution

The correct option is B $(\frac{1}{2})^{20} C_{10}$
$^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + . . . . . . +^{20} C_{10}$

we know that

$(1 + x)^{20} =^{20} C_{0} +^{20} C_{1} x + . . . . . . +^{20} C_{10} x^{10} + . . . . . +^{20} C_{20} x^{2 a}$

Put $x = - 1$ in the above expansion

$0 =^{20} C_{0} -^{20} C_{1} +^{20} C_{2} + . . . . . . .^{20} C_{9} +^{20} C_{10} -^{20} C_{11} + . . . . +^{20} C_{20}$

$0 =^{20} C_{0} -^{20} C_{1} +^{20} C_{2} + . . . . . . +^{20} C_{9} +^{20} C_{10} -^{2} C_{9} +^{2} C_{8} . . . . . . +^{20} C_{0}$

$0 = 2 (^{20} C_{0} -^{20} C_{1} + . . . . . .^{20} C_{9}) +^{20} C_{10}$

$2 (^{20} C_{0} -^{20} C_{1} + . . . . . . +^{20} C_{10}) +^{20} C_{10} +^{20} C_{10} =^{20} C_{10}$

$2 (^{20} C_{0} -^{20} C_{1} + . . . . . . +^{20} C_{10}) =^{20} C_{10}$

$^{20} C_{0} -^{20} C_{1} + . . . . . +^{20} C_{10} = {\frac{1}{2}}^{20} C_{10}$

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Similar questions

^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + . . . . . - . . . . . +^{20} C_{10}

^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + . . . . . - . . . . . . +^{20} C_{10}

{^{20} C}_{0} - {^{20} C}_{1} + {^{20} C}_{2} - {^{20} C}_{3} + . . . . + {^{20} C}_{10}

^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + . . . +^{20} C_{10} =

The sum of the series $^{20} C_{0} +^{20} C_{1} +^{20} C_{2} + . . . +^{20} C_{10}$ is

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