The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1 + \alpha x)^4$ and of $(1 - \alpha x)^6$ is the same if  $\alpha$ equals

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

If the coefficients of $x^{7}$ and $x^{8}$ in the expansion of $\left(2+\frac{x}{3}\right)^{n}$ are equal, then the value of $n$ is equal to $.....$

Let the coefficients of three consecutive terms in the binomial expansion of $(1+2 x)^{ n }$ be in the ratio $2: 5: 8$. Then the coefficient of the term, which is in the middle of these three terms, is $...........$.

The coefficient of $x^8$ in the expansion of $(1 -x^4)^4 (1 + x)^5$ is :-

The coefficient of ${x^5}$ in the expansion of ${(x + 3)^6}$ is