If the sum of the coefficients in the expansion of $(x + y)^n$ is $1024$,then the value of the greatest coefficient in the expansion is

The sum of all possible values of $n \in N$ such that the coefficients of $x$, $x^2$, and $x^3$ in the expansion of $(1+x^2)^2(1+x)^n$ are in arithmetic progression is:

If $ab \neq 0$ and the sum of the coefficients of $x^7$ and $x^4$ in the expansion of $\left(\frac{x^2}{a}-\frac{b}{x}\right)^{11}$ is $0$,then

Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+x)^{99}$. Let $a$ be the middle term in the expansion of $(2+\frac{1}{\sqrt{2}})^{200}$. If $\frac{{}^{200}C_{99} K}{a} = \frac{2^{\ell} m}{n}$,where $m$ and $n$ are odd numbers,then the ordered pair $(\ell, n)$ is equal to:

If $\left(\frac{3^{6}}{4^{4}}\right) k$ is the term independent of $x$ in the binomial expansion of $\left(\frac{x}{4}-\frac{12}{x^{2}}\right)^{12}$,then $k$ is equal to ...... .

The coefficient of $t^4$ in the expansion of ${\left( {\frac{{1 - {t^6}}}{{1 - t}}} \right)^3}$ is