3003010-3013011+…30203030 is equal to

C0 30 C10 30-C1 30 C11 30+......C20 30 C30 30 (1+x)30=C0 30+C1 30 x+C2 30 x2+.........C30 30 x30 1-1x30=C0 30-C1 30 1x+C2 1x2 30+.......C30 30 1x30 (1+x)30 1-1x30=C0 30+C1 30 x+......+C30 30 x30 C0-C1 1x+......+C10 30 1x10-C11 30 1x11 30 30.....+C30 30 1x30Comparing the Coeff of x-10 we get C0 30 C10 30-C1 30 C11 30+.......C20 30 C30 30 =(x2-1)30x30=(1-x2)30x30=C10 30

3003010-3013011+…30203030 is equal to

C0 30 C10 30-C1 30 C11 30+......C20 30 C30 30 (1+x)30=C0 30+C1 30 x+C2 30 x2+.........C30 30 x30 1-1x30=C0 30-C1 30 1x+C2 1x2 30+.......C30 30 1x30 (1+x)30 1-1x30=C0 30+C1 30 x+......+C30 30 x30 C0-C1 1x+......+C10 30 1x10-C11 30 1x11 30 30.....+C30 30 1x30Comparing the Coeff of x-10 we get C0 30 C10 30-C1 30 C11 30+.......C20 30 C30 30 =(x2-1)30x30=(1-x2)30x30=C10 30

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$(\begin{matrix} 30 \\ 0 \end{matrix}) (\begin{matrix} 30 \\ 10 \end{matrix}) - (\begin{matrix} 30 \\ 1 \end{matrix}) (\begin{matrix} 30 \\ 11 \end{matrix}) + \dots (\begin{matrix} 30 \\ 20 \end{matrix}) (\begin{matrix} 30 \\ 30 \end{matrix})$ is equal to

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${}^{30}C_{11}$

${}^{60}C_{10}$

${}^{30}C_{10}$

${}^{65}C_{55}$

answer is C.

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Detailed Solution

${}^{30}{C_{0}} {}^{30}{C_{10}} - {}^{30}{C_{1}} {}^{30}{C_{11}} + . . . . . . {}^{30}{C_{20}} {}^{30}{C_{30}} {(1 + x)}^{30} = {}^{30}{C_{0}} + {}^{30}{C_{1}} x + {}^{30}{C_{2}} x^{2} + . . . . . . . . . {}^{30}{C_{30}} x^{30} {(1 - \frac{1}{x})}^{30} = {}^{30}{C_{0}} - {}^{30}{C_{1}} \frac{1}{x} + {}^{30}{C_{2} \frac{1}{x^{2}}} + . . . . . . . {}^{30}{C_{30}} \frac{1}{x^{30}} {(1 + x)}^{30} {(1 - \frac{1}{x})}^{30} = ({}^{30}{C_{0}} + {}^{30}{C_{1}} x + . . . . . . + {}^{30}{C_{30}} x^{30}) ({}^{30}{C_{0} - {}^{30}{C_{1} \frac{1}{x} + . . . . . . + {}^{30}{C_{10}} \frac{1}{x^{10}} - {}^{30}{C_{11}} \frac{1}{x^{11}}}} . . . . . + {}^{30}{C_{30}} \frac{1}{x^{30}})$

Comparing the Coeff of $x^{- 10}$ we get

${}^{30}{C_{0}} {}^{30}{C_{10}} - {}^{30}{C_{1}} {}^{30}{C_{11}} + . . . . . . . {}^{30}{C_{20}} {}^{30}{C_{30}} = \frac{{(x^{2} - 1)}^{30}}{x^{30}} = \frac{{(1 - x^{2})}^{30}}{x^{30}} = {}^{30}{C_{10}}$