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,$k = (2r+1)-1 = 2r$. Thus,the coef

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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,$k = (2r+1)-1 = 2r$. Thus,the coefficient is ${}^{43}C_{2r}$.For the $(r+2)$-th term,$k = (r+2)-1 = r+1$. Thus,the coefficient is ${}^{43}C_{r+1}$.Given that the coefficients are equal:${}^{43}C_{2r} = {}^{43}C_{r+1}$.Using the property ${}^{n}C_{a} = {}^{n}C_{b} \implies a = b$ or $a+b = n$:Case $1$: $2r = r+1 \implies r = 1$.Case $2$: $2r + (r+1) = 43 \implies 3r + 1 = 43 \implies 3r = 42 \implies r = 14$.Since $r$ must be a positive integer such that the terms exist,we consider $r = 14$ as the standard solution provided in the options.

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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