What is the constant term in the binomial expansion of $(1+3x)^n \left(1+\frac{1}{3x}\right)^n$?

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

Let $\alpha > 0, \beta > 0$ be such that $\alpha^{3} + \beta^{2} = 4$. If the maximum value of the term independent of $x$ in the binomial expansion of $(\alpha x^{\frac{1}{9}} + \beta x^{-\frac{1}{6}})^{10}$ is $10k$,then $k$ is equal to

If the $r$-th and $(r+1)$-th terms in the expansion of $(p+q)^{n}$ are equal,then the value of $\frac{(n+1)q}{r(p+q)}$ is

What is the coefficient of $x^{100}$ in $(1 + x + x^2 + x^3 + \dots + x^{100})^3$?

The term independent of $x$ in the expansion of $\left[\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right]^{10}, x \neq 1,$ is equal to ....... .