Q. The coefficient of $x^{53}$ in the expansion of $$$\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 $\sum_{m = 0}^{100}{\ ^{100}C_{m}}({x - 3)}^{100 - m} \bullet 2^{m}$ can be rewritten as
$\lbrack(x - 3) + 2\rbrack^{100} = (x - 1)^{100} = (1 - x)^{100}$
$\therefore$ $x^{53}$ will occur in $T_{54}$
$\Longrightarrow \ \ T_{54} = \ ^{100}C_{53}( - x)^{53}\$
$\therefore$ Required coefficient is$- \ ^{100}C_{53}.$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.