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# If the sum of the coefficients in the expansion of $(b+c)^{20}[1+(a-2)x]^{20}$ is square of the sum of the coefficients in the expansion $[2bcx-(b+c)y]^{10}$ where $a,b,c$ are positive constants, then $a,b,c$ satisfies the equation?

$\begin{array}{1 1} (A) bc+ac=2bc \\ (B) ab+ac+bc=0 \\ (C) ac+bc=2ab \\ (D) ab+ac=2bc \end{array}$

Can you answer this question?

## 1 Answer

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The sum of the coefficients in the expansion $(b+c)^{20}.[1+(a-2)x]^{20}$
is given by taking $x=1$
$=(b+c)^{20}(a-1)^{20}$ = $S_1$ (say)
Similarly, the sum of the coefficients in the expansion $[2bcx-(b+c)y]^{10}$
is given by taking $x=y=1$
$=[2bc-(b+c)]^{10}$ = $S_2$ (say)
Given: $S_1=S_2^2$
$\Rightarrow\:(b+c)^{20}(a-1)^{20}=(2bc-b-c)^{20}$
$\Rightarrow\:(b+c)(a-1)=2bc-b-c$
$\Rightarrow\:ab+ac=2bc$
answered Oct 23, 2013

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