The sum of the series $^{2020} C_{0} -^{2020} C_{1} +^{2020} C_{2} -^{2020} C_{3} + . . . . . +^{2020} C_{1010}$ is

{\frac{1}{2}}^{2020} C_{1010}

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^{2020} C_{1010}

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Zero

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{\frac{- 1}{2}}^{2020} C_{1010}

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Solution

The correct option is A ${\frac{1}{2}}^{2020} C_{1010}$

$(1 - x)^{m} =^{n} C_{0} -^{n} C_{1} x + \dots +^{n} C_{n} x^{n}$
$Put x = 1 and n = 2020$
$0 =^{2020} C_{0} -^{2020} C_{1} +^{2020} C_{2} + \dots^{2020} C_{2020}$
${also}^{n} C_{r} =^{n} C_{n - r}$
$^{2020} C_{2020} =^{2020} C o$
$[2 [^{2020} C_{0} -^{2020} C_{1} + \dots] +^{2020} C_{1010}] = 0$
$So^{2020} C_{0} -^{2020} C_{1} + \dots -^{2020} C_{1009} =$
${\frac{- 1}{2}}^{2020} C_{1010}$

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