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
$0$
$6$
$4$
$8$
The sum of the binomial coefficients of ${\left[ {2\,x\,\, + \,\,\frac{1}{x}} \right]^n}$ is equal to $256$ . The constant term in the expansion is
Coefficient of ${x^2}$ in the expansion of ${\left( {x - \frac{1}{{2x}}} \right)^8}$ is
The Coefficient of $x ^{-6}$, in the expansion of $\left(\frac{4 x}{5}+\frac{5}{2 x^2}\right)^9$, is $........$.
The term independent of $x$ in ${\left[ {\sqrt{\frac{ x }{3}} + \frac{{\sqrt 3 }}{{{x^2}}}} \right]^{10}}$ is
The expression $[x + (x^3-1)^{1/2}]^5 + [x - (x^3-1)^{1/2}]^5$ is a polynomial of degree :