The coefficient of $x^{7}$ in the expression $(1+x)^{10}+x(1+x)^{9}+x^{2}(1+x)^{8}+\ldots+x^{10}$ is
$120$
$330$
$210$
$420$
If ${x^m}$occurs in the expansion of ${\left( {x + \frac{1}{{{x^2}}}} \right)^{2n}},$ then the coefficient of ${x^m}$ is
The term independent of $x$ in the expression of $\left(1-x^{2}+3 x^{3}\right)\left(\frac{5}{2} x^{3}-\frac{1}{5 x^{2}}\right)^{11}, x \neq 0$ is
In the expansion of $\left(\frac{\mathrm{x}}{\cos \theta}+\frac{1}{\mathrm{x} \sin \theta}\right)^{16},$ if $\ell_{1}$ is the least value of the term independent of $x$ when $\frac{\pi}{8} \leq \theta \leq \frac{\pi}{4}$ and $\ell_{2}$ is the least value of the term independent of $x$ when $\frac{\pi}{16} \leq \theta \leq \frac{\pi}{8},$ then the ratio $\ell_{2}: \ell_{1}$ is equal to
Expand using Binomial Theorem $\left(1+\frac{ x }{2}-\frac{2}{ x }\right)^{4}, x \neq 0$