Q. The coefficient of $x^{100}$ in the expansion of $\sum_{j = 0}^{200}{({1 + x)}^{j}}$ is
A) $\begin{pmatrix}
200 \\
100
\end{pmatrix}$
B) $\begin{pmatrix}
201 \\
102
\end{pmatrix}$
C) $\begin{pmatrix}
200 \\
101
\end{pmatrix}$
D) $\begin{pmatrix}
201 \\
100
\end{pmatrix}$
Explanation
(d)
$\therefore$ coefficient of $x^{100}$ in the expansion of $\sum_{j = 0}^{200}{({1 + x)}^{j}}$ will be $\sum_{j = 0}^{200}{\ \ }j_{C_{100}}$
$\ \ = \left\lbrack \ ^{100}C_{100} + \ ^{101}C_{100} + \ ^{102}C_{100} + \ldots + \ ^{200}C_{100} \right\rbrack$
$\ \ \ \ \left\lbrack \because\ ^{n}C_{n} + \ ^{n + 1}C_{n} + \ ^{n + 2}C_{n} + \ldots + \ ^{2n - 1}C_{n} = \ ^{2n}C_{n + 1} \right\rbrack$
$\ \ \ \ \ \ \ \ \ = \begin{pmatrix}
201 \\
100
\end{pmatrix}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.