Question

The coefficient of $x^{53}$ in ${\sum }_{r=0}^{100}{{}^{100}C}_r{(x-3)}^{100-r}.2^r$ is

• A. ${{}^{100}C}_{47}$
• B. ${{}^{100}C}_{53}$
• C. $-\left({{}^{100}C}_{53}\right)$
• D. ${{}^{100}C}_{100}$

Answer 1

Answer: (C)

$-\left({{}^{100}C}_{53}\right)$

$\sum _{r=0}^{100}{}^{100}{C}_r{\left(x-3\right)}^{100-r}{2}^r$

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

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

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

${=}^{100}{C}_0{x}^{100}{\left(-1\right)}^0{+}^{100}{C}_1{x}^{99}{\left(-1\right)}^1+----{+}^{100}{C}_{47}{\left(x\right)}^{53}{\left(-1\right)}^{47}+--$

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

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

Hence$,$ Option $\left(C\right)$ is correct$.$