# The sum of last $30$ coefficients in the expansion of $(1+x)^{59}$ when expanded in ascending power of $x$ is ?

$\begin{array}{1 1} 2^{29} \\ 2^{28} \\ ^{60} C_{30} -2^{19} \\ 2^{58} \end{array}$

Toolbox:
• $^nC_0+^nC_1+^nC_2+........^nC_n=2^n$
There are 60 terms in the expansion of $(1+x)^{59}$.
Let the sum of last 30 coefficients be $S$
$\therefore\:S=^{59}C_{30}+^{59}C_{31}+........^{59}C_{59}$
Since $^nC_r=^nC_{n-r}$, we can write $S$ as
$S=^{59}C_{29}+^{28}C_{31}+........^{59}C_{0}$
$=^{59}C_0+^{59}C_1+..........^{59}C_{28}+^{59}C_{29}$
Adding both the $S$ we get
$2S=^{59}C_0+^{59}C_1+^{59}C_2+.........^{59}C_{59}$
$\Rightarrow\:2S=2^{59}$
$\therefore\:S=2^{58}$