The value of
$(\begin{matrix} 30 \\ 0 \end{matrix}) (\begin{matrix} 30 \\ 10 \end{matrix}) - (\begin{matrix} 30 \\ 1 \end{matrix}) (\begin{matrix} 30 \\ 11 \end{matrix}) + (\begin{matrix} 30 \\ 2 \end{matrix}) (\begin{matrix} 30 \\ 12 \end{matrix}) - ...... + (\begin{matrix} 30 \\ 20 \end{matrix}) (\begin{matrix} 30 \\ 30 \end{matrix}) =$

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$^{60} C_{20}$

$^{30} C_{10}$

$^{60} C_{30}$

$^{40} C_{30}$

answer is B.

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Detailed Solution

${(1 - x)}^{30} = {}^{30}C_{0} x^{0} -^{30} C_{1} x^{1} +^{30} C_{2} x^{2} + ........ + {(- 1)}^{30} C_{30} x^{30} {(x + 1)}^{30} = 30 c_{0} x^{30} + 30 c_{1} x^{29} + ................. + 30 c_{30}$

Multiplying (1) and (2) and equating the coefficient of $x^{20}$ on both sides, we get required sum is equal to coefficient of $x^{20}$ in ${(1 - x^{2})}^{30},$ which is given by ${}^{30}C_{10} .$