If for positive integers r 1, n 2,the coe | vedclass

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(A) The general term in the expansion of $(1 + x)^{2n}$ is given by $T_{k+1} = ^{2n}C_k x^k$.The coefficient of $x^{3r}$ is $^{2n}C_{3r}$ and the coef

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$2r + 1$

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(A) The general term in the expansion of $(1 + x)^{2n}$ is given by $T_{k+1} = ^{2n}C_k x^k$.The coefficient of $x^{3r}$ is $^{2n}C_{3r}$ and the coefficient of $x^{r+2}$ is $^{2n}C_{r+2}$.Given that the coefficients are equal,we have $^{2n}C_{3r} = ^{2n}C_{r+2}$.Using the property $^{n}C_a = ^{n}C_b \Rightarrow a = b$ or $a + b = n$,we have two cases:Case $1$: $3r = r + 2$ $\Rightarrow 2r = 2$ $\Rightarrow r = 1$. However,the problem states $r > 1$,so this case is rejected.Case $2$: $3r + (r + 2) = 2n$ $\Rightarrow 4r + 2 = 2n$ $\Rightarrow n = 2r + 1$.Thus,$n = 2r + 1$.

If for positive integers $r > 1, n > 2$,the coefficients of the $(3r)^{th}$ and $(r + 2)^{th}$ powers of $x$ in the expansion of $(1 + x)^{2n}$ are equal,then $n$ is equal to

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