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(B) Let $S$ be the sum of the last eight coefficients in the expansion of $(1 + x)^{15}$.$S = {}^{15}C_8 + {}^{15}C_9 + {}^{15}C_{10} + {}^{15}C_{11}

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$2^{14}$

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(B) Let $S$ be the sum of the last eight coefficients in the expansion of $(1 + x)^{15}$.$S = {}^{15}C_8 + {}^{15}C_9 + {}^{15}C_{10} + {}^{15}C_{11} + {}^{15}C_{12} + {}^{15}C_{13} + {}^{15}C_{14} + {}^{15}C_{15}$ ...$(1)$Using the property ${}^{n}C_r = {}^{n}C_{n-r}$,we can write:$S = {}^{15}C_7 + {}^{15}C_6 + {}^{15}C_5 + {}^{15}C_4 + {}^{15}C_3 + {}^{15}C_2 + {}^{15}C_1 + {}^{15}C_0$ ...$(2)$Adding $(1)$ and $(2)$:$2S = ({}^{15}C_0 + {}^{15}C_1 + \ldots + {}^{15}C_7) + ({}^{15}C_8 + {}^{15}C_9 + \ldots + {}^{15}C_{15})$$2S = \sum_{r=0}^{15} {}^{15}C_r = 2^{15}$$S = \frac{2^{15}}{2} = 2^{14}$

The sum of the last eight coefficients in the expansion of $(1 + x)^{15}$ is:

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