If $C_0, C_1, C_2, \ldots, C_8$ are the binomial coefficients in the expansion of $(1+x)^8$,then $\sum_{r=1}^8 r^3 \frac{C_r}{C_{r-1}} =$

The coefficient of $x^7$ in the expansion of $(1 - x - x^2 + x^3)^6$ is

If the coefficient of the middle term in the expansion of $(1 + x)^{2n + 2}$ is $p$ and the coefficients of the two middle terms in the expansion of $(1 + x)^{2n + 1}$ are $q$ and $r$,then:

In the expansion of $(x-1)(x-2) \ldots (x-18)$, the coefficient of $x^{17}$ is:

The term independent of $x(x>0, x \neq 1)$ in the expansion of $\left[\frac{(x+1)}{\left(x^{2 / 3}-x^{1 / 3}+1\right)}-\frac{(x-1)}{(x-\sqrt{x})}\right]^{10}$ is:

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 $(a, b):$