If the sum of the coefficients in the expansion of $(x+y)^{n}$ is $4096,$ then the greatest coefficient in the expansion is .... .

The coefficient of $x^n$ in the expansion of $(1 + x)(1 - x)^n$ is

If the constant term in the binomial expansion of $\left(\frac{x^{5/2}}{2} - \frac{4}{x^{\ell}}\right)^9$ is $-84$ and the coefficient of $x^{-3\ell}$ is $2^{\alpha}\beta$,where $\beta < 0$ is an odd number,then $|\alpha\ell - \beta|$ is equal to

Let $[t]$ denote the greatest integer $\leq t$. If the constant term in the expansion of $\left(3x^2 - \frac{1}{2x^5}\right)^7$ is $\alpha$,then $[\alpha]$ is equal to $............$.

The numerically greatest term in the expansion of $(2x - 3y)^{11}$ when $x = \frac{1}{3}$ and $y = \frac{1}{2}$ is:

Let $a_0, a_1, \ldots, a_{23}$ be real numbers such that $(1+\frac{2}{5} x)^{23} = \sum_{i=0}^{23} a_i x^i$ for every real number $x$. Let $a_r$ be the largest among the numbers $a_j$ for $0 \leq j \leq 23$. Then the value of $r$ is $....$ .