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

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

We have, ${\sum }_{m=0}^{100}{}^{100}{C}_m{(x-3)}^{100-m}.2^m$

$={(x-3)}^{100}{+}^{100}{C}_1{(x-3)}^{99}\times 2^1{+}^{100}{C}_2{(x-3)}^{98}\times 2^2+...{+}^{100}{C}_{100}{2}^{100}={[(x-3)+2]}^{100}={(x-1)}^{100}={(1-x)}^{100}$

$\therefore$ Coefficients of $x^{54}{=}^{100}{C}_{53}{(-1)}^{53}={-}^{100}{C}_{53}$