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The value of $^nC_0.^{2n}C_r$ - $^nC_1^{2-2}C_r$ + $^nC_2.^{2n-4}C_r$ - ...., if $r \lt n$ is

$\begin{array}{1 1} (A) 2^{2n-r}.^nC_{r-n}\\ (B) 2^{2n-r+2}.^nC_{r-n+1}\\ (C) 2^{2n-r}.^nC_{r} \\ (D) 0 \end{array} $

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$\big[(1+x)^2-1\big]^n=^nC_0.(1+x)^{2n}-^nC_1.(1+x)^{2n-2}+^nC_2.(1+x)^{2n-4}-.............$
$\Rightarrow\:x^n(2+x)^n=^nC_0.(1+x)^{2n}-^nC_1.(1+x)^{2n-2}+^nC_2.(1+x)^{2n-4}-.............$
Equating the coefficient of $x^r$ on both the sides,
Coeff. of $x^r\:\: in\:\: x^n(2+x)^n=$ coeff. of $x^{r-n}$ in $(2+x)^n$
$\therefore\:^nC_{r-n}.2^{2n-r}=$$^nC_0^{2n}C_r-^nC_1^{2n-2}C_r+^nC_2^{2n-4}C_r-..........$
But  if  $r<n$,  then  $^nC_{n-r}$  doesnot exist.
$\therefore$ The given series $= 0$
 

 

answered Nov 24, 2013 by rvidyagovindarajan_1
 

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