In the expansion of ${\left( x - \frac{3}{x^2} \right)^9}$,the term independent of $x$ is

Suppose $l, m, n$ respectively represent the coefficient of $x^{10}$,the constant term,and the coefficient of $x^{-10}$ in the expansion of $\left(a x^2+\frac{b}{x^3}\right)^{15}$. If $\frac{l}{m}+\frac{m}{n}=\frac{26}{11}$,then $a^2: b^2=$

Prove that the coefficient of $x^{n}$ in the expansion of $(1+x)^{2n}$ is twice the coefficient of $x^{n}$ in the expansion of $(1+x)^{2n-1}$.

The $r^{th}$ term in the expansion of $(a + 2x)^n$ is

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

If the second,third,and fourth terms in the expansion of $(x + a)^n$ are $240, 720,$ and $1080$ respectively,then the value of $n$ is