The coefficient of ${x^{ - 9}}$ in the expansion of ${\left( {\frac{{{x^2}}}{2} - \frac{2}{x}} \right)^9}$ is

In the expansion of ${(1 + x)^n}$ the coefficient of $p^{th}$ and ${(p + 1)^{th}}$ terms are respectively $p$ and $q$. Then $p + q = $

If ${x^m}$occurs in the expansion of ${\left( {x + \frac{1}{{{x^2}}}} \right)^{2n}},$ then the coefficient of ${x^m}$ is

For a positive integer $n,\left(1+\frac{1}{x}\right)^{n}$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are in the ratio, $2: 5: 12,$ then $n$ is equal to

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 $x ^7$ in $\left(1-x+2 x^3\right)^{10}$ is $........$.