Question

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

• A. $2^{29}$
• B. $2^{28}$
• C. ${{}^{60}C}_{30}-2^{19}$
• D. $2^{58}$

Answer 1

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

$(1+x{)}^{59}={}^{59}{C}_0+{}^{59}{C}_{1}x+{}^{59}{C}_2{x}^2+....{}^{59}{C}_{59}{x}^{59}$
Substituting $x=1$, we get
$2^{59}={}^{59}{C}_0+{}^{59}{C}_1+{}^{59}{C}_2+....{}^{59}{C}_{59}$
$2^{59}=2[{}^{59}{C}_0+{}^{59}{C}_1+{}^{59}{C}_2+....{}^{59}{C}_{28}]+{}^{59}{C}_{29}+{}^{59}{C}_{30}$
$2^{59}=2[{}^{59}{C}_0+{}^{59}{C}_1+{}^{59}{C}_2+....{}^{59}{C}_{28}+{}^{59}{C}_{29}]$
Since ${}^{59}{C}_{30}={}^{59}{C}_{29}$
$2^{59}=2[{}^{59}{C}_{59}+{}^{59}{C}_{58}+{}^{59}{C}_{57}+....{}^{59}{C}_{31}+{}^{59}{C}_{30}]$
$2^{59}=2\left(requiredsum\right)$
Required sum $=2^{58}$