If $A$ and $B$ are coefficients of $x^{n}$ in the expansions of $(1+x)^{2n}$ and $(1+x)^{2n-1}$ respectively,then $A / B$ is equal to

The coefficient of ${x^{39}}$ in the expansion of ${\left( {{x^4} - \frac{1}{{{x^3}}}} \right)^{15}}$ is

If the three consecutive coefficients in the expansion of $(1 + x)^n$ are $28, 56$ and $70,$ then the value of $n$ is

If the constant term in the binomial expansion of $\left(\frac{x^{5/2}}{2} - \frac{4}{x^{\ell}}\right)^9$ is $-84$ and the coefficient of $x^{-3\ell}$ is $2^{\alpha}\beta$,where $\beta < 0$ is an odd number,then $|\alpha\ell - \beta|$ is equal to

The positive integer $k$ for which $\frac{(101)^{k/2}}{k!}$ is a maximum is

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.