The coefficient of $x^4$ in ${\left[ {\frac{x}{2}\,\, - \,\,\frac{3}{{{x^2}}}} \right]^{10}}$ is :
$\frac{{405}}{{256}}$
$\frac{{504}}{{259}}$
$\frac{{450}}{{263}}$
$\frac{{405}}{{512}}$
Find $a$ if the $17^{\text {th }}$ and $18^{\text {th }}$ terms of the expansion ${(2 + a)^{{\rm{50 }}}}$ are equal.
If the constant term in the expansion of $\left(\frac{\sqrt[5]{3}}{x}+\frac{2 x}{\sqrt[3]{5}}\right)^{12}, x \neq 0$, is $\alpha \times 2^8 \times \sqrt[5]{3}$, then $25 \alpha$ is equal to :
The coefficient of $x^{50}$ in the binomial expansion of ${\left( {1 + x} \right)^{1000}} + x{\left( {1 + x} \right)^{999}} + {x^2}{\left( {1 + x} \right)^{998}} + ..... + {x^{1000}}$ is
The greatest value of the term independent of $x$ in the expansion of ${\left( {x\sin \theta + \frac{{\cos \theta }}{x}} \right)^{10}}$ is
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 expansion of $(1+x)^{19}$ is $....$