Question

The coefficient of $ x^{53} $ in the expansion $ \displaystyle\sum^{100}_{m=0} $ $ ^{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,
$\displaystyle\sum_{m=0}^{100}{ }^{100} C_{m}(x-3)^{100-m} 2^{m}=[(x-3)+2]^{100} $
$ {\left[\because \displaystyle\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 $(x-1)^{100}={ }^{100} C_{47}(-1)^{47}$
$={ }^{-100} C_{47}={ }^{-100} C_{53}$