If the coefficients of the three consecutive terms in the expansion of $(1+x)^n$ are in the ratio $1:5:20$,then the coefficient of the fourth term is $............$.

If the coefficients of $r$-th and $(r+1)$-th terms in the expansion of $(3+7x)^{29}$ are equal,then $r$ is equal to

If the coefficients of $x^7$ and $x^8$ in the expansion of $[2 + \frac{x}{3}]^n$ are equal,then the value of $n$ is:

The term independent of '$x$' in the expansion of $\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)^{10}$,where $x \neq 0, 1$,is equal to $.....$

In the expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^n, n \in N$,if the ratio of the $15^{\text{th}}$ term from the beginning to the $15^{\text{th}}$ term from the end is $\frac{1}{6}$,then the value of ${}^n C_3$ is:

The term independent of $x$ in the binomial expansion of $\left( 1 - \frac{1}{x} + 3x^5 \right) \left( 2x^2 - \frac{1}{x} \right)^8$ is