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(A) The general term in the expansion of $(1 + x)^{2n + 1}$ is given by $T_{r+1} = {}^{2n+1}C_r x^r$.The coefficients are the binomial coefficients ${

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$\frac{(2n + 1)!}{n!(n + 1)!}$

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(A) The general term in the expansion of $(1 + x)^{2n + 1}$ is given by $T_{r+1} = {}^{2n+1}C_r x^r$.The coefficients are the binomial coefficients ${}^{2n+1}C_r$ for $r = 0, 1, 2, \dots, 2n+1$.For an expansion $(1+x)^N$,the greatest coefficient is the middle term coefficient. Here $N = 2n + 1$,which is odd.For an odd power $N$,there are two middle terms at $r = \frac{N-1}{2}$ and $r = \frac{N+1}{2}$.Substituting $N = 2n + 1$:$r_1 = \frac{2n+1-1}{2} = n$$r_2 = \frac{2n+1+1}{2} = n+1$The greatest coefficients are ${}^{2n+1}C_n$ and ${}^{2n+1}C_{n+1}$.Calculating ${}^{2n+1}C_n = \frac{(2n+1)!}{n!(2n+1-n)!} = \frac{(2n+1)!}{n!(n+1)!}$.Thus,the greatest coefficient is $\frac{(2n + 1)!}{n!(n + 1)!}$.

The greatest coefficient in the expansion of $(1 + x)^{2n + 1}$ is

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