If the coefficients of the (2r+1)-th term and | vedclass

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

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$14$

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(B) The general term in the expansion of $(1+x)^n$ is given by $T_{k+1} = {}^{n}C_k x^k$. For the $(2r+1)$-th term,$k = (2r+1)-1 = 2r$. The coefficient is ${}^{43}C_{2r}$. For the $(r+2)$-th term,$k = (r+2)-1 = r+1$. The coefficient is ${}^{43}C_{r+1}$. Given that the coefficients are equal: ${}^{43}C_{2r} = {}^{43}C_{r+1}$. Using the property ${}^{n}C_a = {}^{n}C_b$,we have either $a = b$ or $a+b = n$. Case $1$: $2r = r+1 \Rightarrow r = 1$. Case $2$: $2r + (r+1) = 43$ $\Rightarrow 3r + 1 = 43$ $\Rightarrow 3r = 42$ $\Rightarrow r = 14$. Since $14$ is one of the options,$r = 14$ is the correct value.

If the coefficients of the $(2r+1)$-th term and the $(r+2)$-th term in the expansion of $(1+x)^{43}$ are equal,then $r$ is equal to:

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