Question

Sum of last $30$ coefficients in the binomial expansion of $(1 + x)^{59}$ is

• A. $2^{29}$
• B. $2^{59}$
• C. $2^{58}$
• D. $2^{59} - 2^{29}$

Answer 1

Answer: (C)

$2^{58}$

Answer 2

We have, $(1+x)^{59}$ Sum of last 30 coefficient of the binomial expansion $={ }^{59} C_{30}+{ }^{59} C_{31}+\ldots+{ }^{59} C_{59}$ We know that, ${ }^{59} C_{0}+{ }^{59} C_{1}+{ }^{59} C_{2}+\ldots+{ }^{59} C_{59}= 2^{59}$ $\Rightarrow\left({ }^{59} C_{0}+{ }^{59} C_{59}\right)+\left({ }^{59} C_{1}+{ }^{59} C_{58}\right)+...$ $+\left({ }^{59} C_{29}+{ }^{59} C_{30}\right)=2^{59}$ $\Rightarrow 2\left({ }^{59} C_{59}+{ }^{59} C_{58}+...+{ }^{59} C_{31}+{ }^{59} C_{30}\right)=2^{59}$ $\left[\because{ }^{n} C_{r}={ }^{n} C_{n-r}\right]$ $\Rightarrow{ }^{59} C_{30}+{ }^{59} C_{31}+...{ }^{59} C_{39}=\frac{2^{59}}{2}=2^{58}$ $\therefore$ Sum of last 30 coefficient of the binomial expansion $(1+x)^{59}$ is $2^{58}$.