The natural number $m$, for which the coefficient of $x$ in the binomial expansion of $\left( x ^{ m }+\frac{1}{ x ^{2}}\right)^{22}$ is $1540,$ is

For the natural numbers $m, n$, if $(1-y)^{m}(1+y)^{n}=1+a_{1} y+a_{2} y^{2}+\ldots .+a_{m+n} y^{m+n}$ and $a_{1}=a_{2}$ $=10$, then the value of $(m+n)$ is equal to:

If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^{\ell}}\right)^9$ is $-84$ and the Coefficient of $x^{-3 \ell}$ is $2^\alpha \beta$, where $\beta < 0$ is an odd number, Then $|\alpha \ell-\beta|$ is equal to

If $\frac{{{T_2}}}{{{T_3}}}$ in the expansion of ${(a + b)^n}$ and $\frac{{{T_3}}}{{{T_4}}}$ in the expansion of ${(a + b)^{n + 3}}$ are equal, then $n=$

The term independent of $x$ in expansion of ${\left( {\frac{{x + 1}}{{{x^{2/3}} - {x^{\frac{1}{3}}} + 1\;}}--\frac{{x - 1}}{{x - {x^{1/2}}}}} \right)^{10}}$ is

Let for the $9^{\text {th }}$ term in the binomial expansion of $(3+6 x)^{n}$, in the increasing powers of $6 x$, to be the greatest for $x=\frac{3}{2}$, the least value of $n$ is $n_{0}$. If $k$ is the ratio of the coefficient of $x ^{6}$ to the coefficient of $x ^{3}$, then $k + n _{0}$ is equal to.