The coefficient of $x^{70}$ in $x^2(1+x)^{98} + x^3(1+x)^{97} + x^4(1+x)^{96} + \ldots + x^{54}(1+x)^{46}$ is ${}^{99}C_p - {}^{46}C_q$. Then a possible value of $p+q$ 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.

If $p$ and $q$ are respectively the coefficients of $x^{-3}$ and $x^{-5}$ in the expansion of $\left(x^{1/3} + \frac{1}{2x^{1/3}}\right)^{21}, x > 0$,then $\frac{5p}{4q} = $

The coefficient of the middle term in the expansion of $(1 + x)^{10}$ is

The second,third and fourth terms in the binomial expansion $(x+a)^n$ are $240, 720$ and $1080$ respectively. Find $x, a$ and $n$.

If the constant term in the binomial expansion of $\left(2x^{r} + \frac{1}{x^{2}}\right)^{10}$ is $180$,then $r$ is equal to $......$