Question

Sum of the last $30$ coefficients in the expansion of ${(1+x)}^{59}$, when expanded in ascending power of $x$ is

• A. $2^{59}$
• B. $2^{58}$
• C. $2^{30}$
• D. $2^{29}$

Answer 1

The correct option is B $2^{58}$

${(1+x)}^n{=}^n{C}_0{+}^n{C}_{1}x{+}^n{C}_2{x}^2........{.}^n{C}_n{x}^n{\Rightarrow }^n{C}_0{+}^n{C}_1{+}^n{C}_2........{.}^n{C}_n=2^n$

For the given expansion ${(1+x)}^{59}$ let sum of last $30$ coefficients be $S$

$S{=}^{59}{C}_{30}{+}^{59}{C}_{31}{+}^{59}{C}_{32}.........{.}^{59}{C}_{59}$ ....(i)

As ${}^n{C}_r{=}^n{C}_{n-r}$

$\Rightarrow S{=}^{59}{C}_{29}{+}^{59}{C}_{28}{+}^{59}{C}_{27}.........{.}^{59}{C}_0$ ....(ii)

Adding (i) and (ii), we get

$\Rightarrow 2S{=}^{59}{C}_0{+}^{59}{C}_1{+}^{59}{C}_2.......{.}^{59}{C}_{59}\Rightarrow 2S=2^{59}\Rightarrow S=2^{58}$

So, option B is correct.