The term independent of $x$ in the expansion of ${(1 + x)^n}{\left( {1 + \frac{1}{x}} \right)^n}$ is
$C_0^2 + 2C_1^2 + .... + (n + 1)C_n^2$
${({C_0} + {C_1} + .... + {C_n})^2}$
$C_0^2 + C_1^2 + ..... + C_n^2$
None of these
Let $\alpha > 0$, be the smallest number such that the expansion of $\left(x^{\frac{2}{3}}+\frac{2}{x^3}\right)^{30}$ has a term $\beta x^{-\alpha}, \beta \in N$. Then $\alpha$ is equal to $.............$.
In the expansion of ${\left( {x - \frac{1}{x}} \right)^6}$, the constant term is
The total number or irrational terms in the binomial expansion of $\left( {{7^{1/5}} - {3^{1/10}}} \right)^{60}$ is
If the $6^{th}$ term in the expansion of the binomial ${\left[ {\frac{1}{{{x^{\frac{8}{3}}}}}\,\, + \,\,{x^2}\,{{\log }_{10}}\,x} \right]^8}$ is $5600$, then $x$ equals to
If ${x^m}$occurs in the expansion of ${\left( {x + \frac{1}{{{x^2}}}} \right)^{2n}},$ then the coefficient of ${x^m}$ is