If the coefficient of $x^7$ in $(ax - \frac{1}{bx^2})^{13}$ and the coefficient of $x^{-5}$ in $(ax + \frac{1}{bx^2})^{13}$ are equal,then $a^4 b^4$ is equal to :

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 $....$ .

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

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

If $C_j$ stands for ${}^nC_j$,then $\frac{C_1}{C_0} + \frac{2 \times C_2}{C_1} + \frac{3 \times C_3}{C_2} + \ldots + \frac{n \times C_n}{C_{n-1}} = $

The number of irrational terms in the expansion of $(3^{\frac{1}{8}}+5^{\frac{1}{4}})^{84}$ is