The coefficient of $x^3$ in the expansion of $(x^2 - x + 1)^{10} (x^2 + 1)^{15}$ is equal to:

The number of positive integers $k$ such that the constant term in the binomial expansion of $\left(2x^3 + \frac{3}{x^k}\right)^{12}, x \neq 0$ is $2^8 \cdot \ell$,where $\ell$ is an odd integer,is:

The coefficient of $x^5$ in the expansion of $(1+x)^{21}+(1+x)^{22}+\ldots+(1+x)^{30}$ is

The coefficient of $x^5$ in the expansion of $(1+x^2)^5(1+x)^4$ is:

The interval in which $x$ must lie so that the numerically greatest term in the expansion of $(1 - x)^{21}$ has the numerically greatest coefficient is

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 $(\alpha x^{\frac{1}{9}} + \beta x^{-\frac{1}{6}})^{10}$ is $10k$,then $k$ is equal to