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The sum of the last eight coefficients in the expansion of $(1+x)^{15}$ is
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The correct answer is:
$2^{14}$
We have ${ }^{15} C_0+{ }^{15} C_1+\ldots+{ }^{15} C_{15}=2^{15}$
$\begin{aligned} & \Rightarrow 2\left({ }^{15} C_8+{ }^{15} C_9+\ldots+{ }^{15} C_{15}\right)=2^{15} \quad\left({ }^n C_r={ }^n C_{n-r}\right) \\
& \Rightarrow{ }^{15} C_8+{ }^{15} C_9+\ldots+{ }^{15} C_{15}=2^{14} \end{aligned}$
$\begin{aligned} & \Rightarrow 2\left({ }^{15} C_8+{ }^{15} C_9+\ldots+{ }^{15} C_{15}\right)=2^{15} \quad\left({ }^n C_r={ }^n C_{n-r}\right) \\
& \Rightarrow{ }^{15} C_8+{ }^{15} C_9+\ldots+{ }^{15} C_{15}=2^{14} \end{aligned}$
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