Question
The coefficient of $$x^{100}$$ in the expansion of $$\sum\limits_{j = 0}^{200} {{{\left( {1 + x} \right)}^j}} $$ is :
A.
\[\left( {\begin{array}{*{20}{c}}
{200}\\
{100}
\end{array}} \right)\]
B.
\[\left( {\begin{array}{*{20}{c}}
{201}\\
{102}
\end{array}} \right)\]
C.
\[\left( {\begin{array}{*{20}{c}}
{200}\\
{101}
\end{array}} \right)\]
D.
\[\left( {\begin{array}{*{20}{c}}
{201}\\
{100}
\end{array}} \right)\]
Answer :
\[\left( {\begin{array}{*{20}{c}}
{200}\\
{100}
\end{array}} \right)\]
Solution :
$${\left( {1 + x} \right)^j} = 1 + {\,^j}{C_1}x + {\,^j}{C_2}{x^2} + {\,^j}{C_3}{x^3} + ..... + {\,^j}{C_{100}}{x^{100}} + ..... + {\,^j}{C_{200}}{x^{200}}$$
$$\therefore $$ Coefficient of $$x^{100}$$ in the expansion of $${\left( {1 + x} \right)^j} = {\,^j}{C_{100}}$$
Coefficient of $$x^{100}$$ in the expansion of
$$\sum\limits_{j = 0}^{200} {{{\left( {1 + x} \right)}^j}} $$ will be equal to $$\sum\limits_{j = 100}^{200} {^j{C_{100}}} $$
$$ = {\,^{100}}{C_{100}} + {\,^{101}}{C_{100}} + {\,^{102}}{C_{100}} + ..... + {\,^{200}}{C_{100}}$$
\[ = {\,^{200}}{C_{100}} = \left( {\begin{array}{*{20}{c}}
{200}\\
{100}
\end{array}} \right)\]