Question

The coefficient of $x^{53}$ in the expansion $\sum _{m=0}^{100}$ ${}^{100}{C}_m(x-3{)}^{100-m}.2^m$ is

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

Answer 1

Answer: (C)

We have,
$\sum _{m=0}^{100}{}^{100}{C}_m(x-3{)}^{100-m}{2}^m=[(x-3)+2{]}^{100}$
$\left[\because \sum _{r=0}^n{}^n{C}_r{x}^{n-r}{y}^r=(x+y{)}^n\right]$
$=(x-1{)}^{100}$
$\therefore T_{r+1}={}^{100}{C}_r{x}^{100-r}(-1{)}^r$
For coefficient of $x^{53},r=47$.
$\therefore$ Coefficient of $x^{53}$ in $(x-1{)}^{100}={}^{100}{C}_{47}(-1{)}^{47}$
$={}^{-100}{C}_{47}={}^{-100}{C}_{53}$