Question

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

• A. ${}^{-100}{C}_{47}$
• B. ${-}^{100}{C}_{53}$
• C. ${}^{65}{C}_{53}$
• D. ${}^{100}{C}_{65}$

Answer 1

The correct option is A ${}^{-100}{C}_{47}$

$\sum _{m=0}^{100}{}^{100}{C}_m{\left(x-3\right)}^{100-m_{2}m}$

$={}^{100}{C}_0{\left(x-3\right)}^{100}+{}^{100}{C}_1{\left(x-3\right)}^{99}.2+.......+{}^{100}{C}_{\infty }{\left(x-3\right)}^{0_{2}100}$

$={\left(x-3+2\right)}^{100}={\left(x-1\right)}^{100}$

Coefficient of $x^{53}$ is ${\left(x-1\right)}^{100}$

$={}^{100}{C}_r{x}^{100-r}{\left(-1\right)}^r$

$r=47$

$={}^{100}{C}_{47}{x}^{53}{\left(-1\right)}^{47}$

$=-{}^{100}{C}_{47}{x}^{53}$

$=-{}^{100}{C}_{47}$