If the coefficients of the (2r + 4)^{th} and | vedclass

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(D) The general term $T_{k+1}$ in the expansion of $(1 + x)^n$ is given by $T_{k+1} = ^nC_k x^k$.For the $(2r + 4)^{th}$ term,$k = (2r + 4) - 1 = 2r +

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$6$

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(D) The general term $T_{k+1}$ in the expansion of $(1 + x)^n$ is given by $T_{k+1} = ^nC_k x^k$.For the $(2r + 4)^{th}$ term,$k = (2r + 4) - 1 = 2r + 3$. The coefficient is $^{18}C_{2r+3}$.For the $(r - 2)^{th}$ term,$k = (r - 2) - 1 = r - 3$. The coefficient is $^{18}C_{r-3}$.Given that the coefficients are equal: $^{18}C_{2r+3} = ^{18}C_{r-3}$.Using the property $^nC_a = ^nC_b$,we have either $a = b$ or $a + b = n$.Case $1$: $2r + 3 = r - 3 \Rightarrow r = -6$ (Not possible as $r$ must be positive).Case $2$: $(2r + 3) + (r - 3) = 18$ $\Rightarrow 3r = 18$ $\Rightarrow r = 6$.Thus,$r = 6$.

If the coefficients of the $(2r + 4)^{th}$ and $(r - 2)^{th}$ terms in the expansion of $(1 + x)^{18}$ are equal,then $r =$

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