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Questions  >>  CBSE XI  >>  Math  >>  Binomial Theorem
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Q)

Find the value of r if the coefficient of $(2r+4)^{th}$ and $(r-2)^{th}$ terms in the expansion of $(1+x)^{18}$ are equal.

$\begin{array}{1 1}(A)\;4\\(B)\;5\\(C)\;6\\(D)\;7\end{array} $

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A)
Toolbox:
  • General term in the expansion of $(a+b)^n$ are $T_{r+1}=nC_ra^{n-r}b^r$
$T_{r+1}=nC_r1^{n-r}x^r$
$\Rightarrow 10C_r (1) ^{18-r}(x)^r$
The coefficient of $(2r+4)^{th}$ term in $(1+x)^{18}=18C_{2r+3}$
The coefficient of $(r-2)^{th}$ term $=18C_{r-3}$
Given :
Coefficient of $(2r+4)^{th}=(r-2)^{th}$ term
$2r+3=r-3$
$2r-r=-3-3$
$r=-6$
Or $2r+3+r-3=18$
$3r=18$
$r=6$
Hence (C) is the correct answer.
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