If the coefficients of the $(2r + 3)^{th}$ and $(r - 1)^{th}$ terms in the expansion of $(1 + x)^{15}$ are equal,then the value of $r$ is:

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

Find the $4^{\text{th}}$ term in the expansion of $(x-2y)^{12}$. (in $x^9y^3$)

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.

If three consecutive coefficients in the binomial expansion of $(x + 1)^n$ in powers of $x$ are in the ratio $2 : 15 : 70$,then the average of these three coefficients is

The ratio of the $5^{th}$ term from the beginning to the $5^{th}$ term from the end in the binomial expansion of $\left( 2^{1/3} + \frac{1}{2(3)^{1/3}} \right)^{10}$ is