Find the mean of the values $0, 1, 2, \dots, n$ with respective weights $^nC_0, ^nC_1, ^nC_2, \dots, ^nC_n$.

Let the coefficients of the middle terms in the expansion of $\left(\frac{1}{\sqrt{6}}+\beta x\right)^{4}$,$(1-3 \beta x)^{2}$ and $\left(1-\frac{\beta}{2} x\right)^{6}$,where $\beta > 0$,respectively form the first three terms of an $A.P.$ If $d$ is the common difference of this $A.P.$,then $50-\frac{2 d}{\beta^{2}}$ is equal to.

In the binomial expansion of $(1+x)^{2k}$,if its middle term is the only numerically greatest term,then $x$ lies in the interval

If $\left(\frac{3^{6}}{4^{4}}\right) k$ is the term independent of $x$ in the binomial expansion of $\left(\frac{x}{4}-\frac{12}{x^{2}}\right)^{12}$,then $k$ is equal to ...... .

If the non-zero coefficient of the $(2r + 4)^{th}$ term is greater than the non-zero coefficient of the $(r - 2)^{th}$ term in the expansion of $(1 + x)^{18}$,then the number of possible integral values of $r$ is:

If the coefficients of $(2 \alpha+4)$-th and $(\alpha-2)$-th terms in the expansion of $(1+x)^{2018}$ are equal,then $\alpha=$