If in the expansion of $(1 + x)^{21}$,the coefficients of $x^r$ and $x^{r + 1}$ are equal,then $r$ is equal to

The coefficient of $x^7$ in the expansion of $\left( \frac{x^2}{2} - \frac{2}{x} \right)^8$ is

The coefficient of $t^{12}$ in the expansion of $(1 + t^2)^6(1 + t^6)(1 + t^{12})$ is:

If the coefficient of the $3^{\text{rd}}$ term from the beginning in the expansion of $\left(ax^2 - \frac{8}{bx}\right)^9$ is equal to the coefficient of the $3^{\text{rd}}$ term from the end in the expansion of $\left(ax - \frac{2}{bx^2}\right)^9$,then the relation between $a$ and $b$ is:

The coefficient of $x^{n-6}$ in the expansion $n! \left[ x - \left( \frac{^nC_0 + ^nC_1}{^nC_0} \right) \right] \left[ \frac{x}{2} - \left( \frac{^nC_1 + ^nC_2}{^nC_1} \right) \right] \left[ \frac{x}{3} - \left( \frac{^nC_2 + ^nC_3}{^nC_2} \right) \right] \dots \left[ \frac{x}{n} - \left( \frac{^nC_{n-1} + ^nC_n}{^nC_{n-1}} \right) \right]$ is equal to:

The sum of all those terms which are rational numbers in the expansion of $(2^{1/3} + 3^{1/4})^{12}$ is: