The sum of the coefficients in the expansion of $(x + y)^n$ is $4096$. The greatest coefficient in the expansion is

The coefficient of $x^4$ in the expansion of $\frac{(1-3 x)^2}{(1-2 x)}$ is equal to

The numerically greatest term in the expansion of $(x+3y)^{13}$,when $x=\frac{1}{2}$ and $y=\frac{1}{3}$ is

Find the coefficient of $\alpha^t$ in the expansion of $(\alpha + p)^{m - 1} + (\alpha + p)^{m - 2} (\alpha + q) + (\alpha + p)^{m - 3} (\alpha + q)^2 + \dots + (\alpha + q)^{m - 1}$,where $\alpha \neq -q$ and $p \neq q$.

If the coefficient of $x^{15}$ in the expansion of $(ax^3 + \frac{1}{bx^{1/3}})^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $(ax^{1/3} - \frac{1}{bx^3})^{15}$,where $a$ and $b$ are positive real numbers,then for each such ordered pair $(a, b):$

Find the coefficient of $x^{15}$ in the product $(1 - x) (1 - 2x) (1 - 2^2 x) (1 - 2^3 x) \dots (1 - 2^{15} x)$.