If the coefficients of $x^7$ and $x^8$ in $(2 + \frac{x}{3})^n$ are equal,then $n$ is

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:

Let $\alpha > 0, \beta > 0$ be such that $\alpha^{3} + \beta^{2} = 4$. If the maximum value of the term independent of $x$ in the binomial expansion of $(\alpha x^{\frac{1}{9}} + \beta x^{-\frac{1}{6}})^{10}$ is $10k$,then $k$ is equal to

What is the coefficient of $x^{100}$ in $(1 + x + x^2 + x^3 + \dots + x^{100})^3$?

If the sum of the coefficients in the expansion of $(x + y)^n$ is $1024$,then the value of the greatest coefficient in the expansion is

If the $k^{\text{th}}$ term in the expansion of $\left(\frac{3}{2} x^2 - \frac{1}{3x}\right)^6$ is independent of $x$,then the numerically greatest term in the expansion of $\left(\frac{3}{2} x^2 - \frac{1}{3x}\right)^k$ when $x = \frac{2}{3}$ is: