Question

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

• A. ${}^{100}{c}_{53}$
• B. ${}^{98}{c}_{53}$
• C. ${}^{65}{c}_{53}$
• D. ${}^{100}{c}_{65}$

Answer 1

The correct option is B ${}^{98}{c}_{53}$

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

${}^{100}{C}_m{(x-3)}^{100-m}{2}^m$ is general term of ${(x-3+2)}^{100}$

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

${(x-1)}^{100}={(1-x)}^{100}$

Hence coefficient of $x^{53}$ in

${(1-x)}^{100}={(-1)}^{53}{}^{100}{C}_{53}={-}^{100}{C}_{53}$