The coefficient of ${x^{100}}$ in the expansion of $\sum\limits_{j = 0}^{200} {{{(1 + x)}^j}} $ is
$\left( \begin{array}{l}200\\100\end{array} \right)$
$\left( \begin{array}{l}201\\102\end{array} \right)$
$\left( \begin{array}{l}200\\101\end{array} \right)$
$\left( \begin{array}{l}201\\100\end{array} \right)$
If the coefficients of the three consecutive terms in the expansion of $(1+ x )^{ n }$ are in the ratio $1: 5: 20$, then the coefficient of the fourth term is $............$.
Coefficient of $x^3$ in the expansion of $(x^2 - x + 1)^{10} (x^2 + 1 )^{15}$ is equal to
The coefficient of ${x^5}$ in the expansion of ${({x^2} - x - 2)^5}$ is
Let the sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^2}\right)^n, x \neq 0, n \in N$, be $376$. Then the coefficient of $x^4$ is $......$
If the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{k}{x^{2}}\right)^{10}$ is $405,$ then $|k|$ equals