Find the coefficient of $x^{6} y^{3}$ in the expansion of $(x+2 y)^{9}$.

For $n \in N$,in the expansion of $\left(\sqrt[4]{x^{-3}}+a \sqrt[4]{x^5}\right)^n$,the sum of all binomial coefficients lies between $200$ and $400$ and the term independent of $x$ is $448$. Then the value of $a$ is

The number of integral terms in the expansion of $\{7^{(1/2)} + 11^{(1/6)}\}^{824}$ is equal to ...................

If the constant term in the binomial expansion of $\left(\frac{x^{5/2}}{2} - \frac{4}{x^{\ell}}\right)^9$ is $-84$ and the coefficient of $x^{-3\ell}$ is $2^{\alpha}\beta$,where $\beta < 0$ is an odd number,then $|\alpha\ell - \beta|$ is equal to

The coefficient of $x^{-7}$ in the expansion of $\left( ax - \frac{1}{bx^2} \right)^{11}$ is

The largest term in the expansion of $(3 + 2x)^{50}$ where $x = \frac{1}{5}$ is