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

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(^{10} C_{5})^{2}

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-^{10} C_{5}

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2^{10} C_{5}

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Solution

The correct option is D $-^{10} C_{5}$
$^{10} C_{0}^{2} -^{10} C_{1}^{2} +^{10} C_{2}^{2} - . . . . . . .^{10} C_{10}^{10} C_{10}$
$(1 + n)^{10} =^{10} C_{0} +^{10} C_{1} n +^{10} C_{2} n^{2} . . . .$
$(1 - n)^{10} =^{10} C_{0} +^{10} C_{1} n +^{10} C_{2} n^{2} . . . .$
$(1 + n)^{10} (1 - n)^{10} =$ coeffi. of $n^{10}$
coeff of $n^{10} = (1 - n^{2})^{10} =^{10} C_{5} (- 1)^{5}$
$= -^{10} C_{5}$

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