Question

The Coefficient of $x^{53}$ in ${\sum }_{m=0}^{100}{C_m}^{100}{(x-3)}^{100-m}{2}^m$ is

• A. $=-{}^{100}{C}_{53}$
• B. $=-{}^{101}{C}_{53}$
• C. $={}^{101}{C}_{47}$
• D. $=-{}^{100}{C}_{47}$

Answer 1

The correct option is A $=-{}^{100}{C}_{53}$

The general term is
$G$
$={}^{100}{C}_m(x-3{)}^{100-m}{2}^m$
This is the general term of
$\left(\right(x-3)+2{)}^{100}$
$=(x-1{)}^{100}$
$=(1-x{)}^{100}$
Hence the coefficient of $x^{53}$ is
$T_{54}$
$=(-1{)}^{53}{}^{100}{C}_{53}$
$=-{}^{100}{C}_{53}$