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

The middle term in the expansion of $(1 + x)^{2n}$ is

If the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{k}{x^{2}}\right)^{10}$ is $405$,then $|k|$ equals

If $21^{\text{st}}$ and $22^{\text{nd}}$ terms in the expansion of $(1+x)^{44}$ are equal,then $x$ 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 total number of irrational terms in the binomial expansion of $(7^{1/5} - 3^{1/10})^{60}$ is: