Question

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

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

Answer 1

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

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

Above expansion can be rewritten as

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

$\therefore x^{53}$ will occur in $T_{54}$ i.e. ${54}^{th}$ term.

So, $m=53$

$\Rightarrow T_{54}={}^{100}{C}_{53}(-x{)}^{53}$

$\therefore$ Required coefficient is ${}^{-100}{C}_{53}$.