The coefficient of $x^8$ in the expansion of $(1 - x^4)^4 (1 + x)^5$ is :-

The number of positive integers $k$ such that the constant term in the binomial expansion of $\left(2x^3 + \frac{3}{x^k}\right)^{12}, x \neq 0$ is $2^8 \cdot \ell$,where $\ell$ is an odd integer,is:

If the coefficients of $x^3$ and $x^4$ in the expansion of $(1 + ax + bx^2)(1 - 2x)^{18}$ in powers of $x$ are both zero,then $(a, b)$ is equal to

Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^5$ in the expansion of $(ax^2 + \frac{70}{27bx})^4$ is equal to the coefficient of $x^{-5}$ in the expansion of $(ax - \frac{1}{bx^2})^7$,then the value of $2b$ is

If the coefficients of the $(2r + 3)^{th}$ and $(r - 1)^{th}$ terms in the expansion of $(1 + x)^{15}$ are equal,then the value of $r$ is:

The term independent of $x$ in the expansion of ${\left( {\frac{1}{2}{x^{1/3}} + {x^{ - 1/5}}} \right)^8}$ is: