Q. The coefficient of $x^{53}$ in the following expansion $\sum_{m = 0}^{100\ }{\ ^{100}C_{m}{(x - 3)}^{100 - m} \bullet 2^{m}}$ is
A) $\ ^{100}C_{47}$
B) $\ ^{100}C_{53}$
C) $- \ ^{100}C_{53}$
D) $- \ ^{100}C_{100}$
Explanation
(c)
The given sigma expansion
$\sum_{m = 0}^{100}{\ ^{100}C_{m}{(x - 3)}^{100 - m}}.2^{m}$ can be written as
$\left\lbrack (x - 3) + 2 \right\rbrack^{100} = (x - 1)^{100} = (x - 1)^{100}$
$\therefore$ Coefficient of $x^{53}$ in ${(1 - x)}^{100} = {( - 1)}^{53}\ ^{100}C_{53} = - \ ^{100}C_{53}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.