The Coefficient of $x ^{-6}$, in the expansion of $\left(\frac{4 x}{5}+\frac{5}{2 x^2}\right)^9$, is $........$.
$5041$
$5042$
$5043$
$5040$
Let $\alpha > 0$, be the smallest number such that the expansion of $\left(x^{\frac{2}{3}}+\frac{2}{x^3}\right)^{30}$ has a term $\beta x^{-\alpha}, \beta \in N$. Then $\alpha$ is equal to $.............$.
If the coefficient of the middle term in the expansion of ${(1 + x)^{2n + 2}}$ is $p$ and the coefficients of middle terms in the expansion of ${(1 + x)^{2n + 1}}$ are $q$ and $r$, then
Find the $r^{\text {th }}$ term from the end in the expansion of $(x+a)^{n}$
If ${x^m}$occurs in the expansion of ${\left( {x + \frac{1}{{{x^2}}}} \right)^{2n}},$ then the coefficient of ${x^m}$ is
In the expansion of $(1 + x)^{43}$ if the co-efficients of the $(2r + 1)^{th}$ and the $(r + 2)^{th}$ terms are equal, the value of $r$ is :