In the expansion of ${\left( \frac{3x^2}{2} - \frac{1}{3x} \right)^9}$,the term independent of $x$ is

The coefficient of $x^{91}$ in the series $^{100}C_1 \cdot 2^8 \cdot (1 - x)^{99} + ^{100}C_2 \cdot 2^7 \cdot (1 - x)^{98} + ^{100}C_3 \cdot 2^6 \cdot (1 - x)^{97} + \dots + ^{100}C_9 \cdot (1 - x)^{91}$ 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 $....$ .

The coefficient of $x^4$ in $\left[ \frac{x}{2} - \frac{3}{x^2} \right]^{10}$ is:

The $16^{th}$ term in the expansion of $(\sqrt{x} - \sqrt{y})^{17}$ is

The number of irrational terms in the binomial expansion of $(3^{1/5} + 7^{1/3})^{100}$ is