In the binomial expansion of $(1+x)^{15}$,the coefficients of $x^{r}$ and $x^{r+3}$ are equal. Then,$r$ 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

The coefficient of $\frac{1}{x}$ in the expansion of ${\left( {1 + x} \right)^n}{\left( {1 + \frac{1}{x}} \right)^n}$ is :-

If the ratio of the $7^{th}$ term from the beginning to the $7^{th}$ term from the end in the expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^n$ is $\frac{1}{6}$,then $n=$

Find the coefficients of $x^r$ where $0 \le r \le (n - 1)$ in the expansion of $(x + 3)^{n - 1} + (x + 3)^{n - 2}(x + 2) + (x + 3)^{n - 3}(x + 2)^2 + ... + (x + 2)^{n - 1}$.

If the coefficients of the $(2r + 1)^{th}$ term and the $(r + 2)^{th}$ term are equal in the expansion of $(1 + x)^{43}$,then the value of $r$ is: