In the expansion of ${\left( {3x - \frac{1}{{{x^2}}}} \right)^{10}}$,the $5^{th}$ term from the end is:

The coefficient of $x^{7}$ in the expression $(1+x)^{10}+x(1+x)^{9}+x^{2}(1+x)^{8}+\ldots+x^{10}$ is

Find the mean of the values $0, 1, 2, \dots, n$ with respective weights $^nC_0, ^nC_1, ^nC_2, \dots, ^nC_n$.

Let the coefficients of $x^{-1}$ and $x^{-3}$ in the expansion of $(2x^{1/5} - x^{-1/5})^{15}$,$x > 0$,be $m$ and $n$ respectively. If $r$ is a positive integer such that $mn^2 = {}^{15}C_r \cdot 2^r$,then the value of $r$ is equal to

$x^r$ occurs in the expansion of ${\left( x^3 + \frac{1}{x^4} \right)^n}$ provided -

The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in the expansion of $(1+x)^{19}$ is $....$