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
$176$
$336$
$352$
$84$
${6^{th}}$ term in expansion of ${\left( {2{x^2} - \frac{1}{{3{x^2}}}} \right)^{10}}$ is
The number of positive integers $k$ such that the constant term in the binomial expansion of $\left(2 x^{3}+\frac{3}{x^{k}}\right)^{12}, x \neq 0$ is $2^{8} \cdot \ell$, where $\ell$ is an odd integer, is......
The coefficient of ${x^n}$in expansion of $(1 + x)\,{(1 - x)^n}$ is
In the expansion of ${(1 + x + {x^3} + {x^4})^{10}},$ the coefficient of ${x^4}$ is
Let the sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^2}\right)^n, x \neq 0, n \in N$, be $376$. Then the coefficient of $x^4$ is $......$