A) \[\left( \begin{align} & 200 \\ & 100 \\ \end{align} \right)\]
B) \[\left( \begin{align} & 201 \\ & 102 \\ \end{align} \right)\]
C) \[\left( \begin{align} & 200 \\ & 101 \\ \end{align} \right)\]
D) \[\left( \begin{align} & 201 \\ & 100 \\ \end{align} \right)\]
Correct Answer: A
Solution :
[a] \[{{(1+x)}^{j}}=1+{{\,}^{j}}{{C}_{1}}x+{{\,}^{j}}{{C}_{2}}{{x}^{2}}+{{\,}^{j}}{{C}_{3}}{{x}^{3}}+.....\] \[{{+}^{j}}{{C}_{100}}{{x}^{100}}+......+{{\,}^{j}}{{C}_{200}}{{x}^{200}}\] \[\therefore \] Coefficient of \[{{x}^{100}}\] in the expansion of \[{{(1+x)}^{j}}={{\,}^{j}}{{C}_{100}}\] Coefficient of \[{{x}^{100}}\] in the expansion of \[\sum\limits_{j=0}^{200}{{{(1+x)}^{j}}}\] will be equal to \[\sum\limits_{j=100}^{200}{^{j}{{C}_{100}}}\] \[={{\,}^{100}}{{C}_{100}}+{{\,}^{101}}{{C}_{100}}+{{\,}^{102}}{{C}_{100}}+....+{{\,}^{200}}{{C}_{100}}\] \[={{\,}^{200}}{{C}_{100}}=\left( \begin{matrix} 200 \\ 100 \\ \end{matrix} \right)\]
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