The sum of the real values of $x$ for which the middle term in the binomial expansion of ${\left( {\frac{{{x^3}}}{3} + \frac{3}{x}} \right)^8}$ equals $5670$ is

In the expansion of ${\left( {x - \frac{1}{x}} \right)^6}$, the constant term is

The coefficient of ${x^{100}}$ in the expansion of $\sum\limits_{j = 0}^{200} {{{(1 + x)}^j}} $ is

$x^r$ occurs in the expansion of ${\left( {{x^3} + \frac{1}{{{x^4}}}} \right)^n}$ provided -

If the ${(r + 1)^{th}}$ term in the expansion of ${\left( {\sqrt[3]{{\frac{a}{{\sqrt b }}}} + \sqrt {\frac{b}{{\sqrt[3]{a}}}} } \right)^{21}}$ has the same power of $a$ and $b$, then the value of $r$ 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 $\left(\alpha x^{\frac{1}{9}}+\beta x^{-\frac{1}{6}}\right)^{10}$ is $10 k$ then $\mathrm{k}$ is equal to