If the term without $x$ in the expansion of $\left( x ^{\frac{2}{3}}+\frac{\alpha}{ x ^3}\right)^{22}$ is $7315$ , then $|\alpha|$ is equal to $...........$.

The coefficient of the middle term in the binomial expansion in powers of $x$ of ${(1 + \alpha x)^4}$ and of ${(1 - \alpha x)^6}$ is the same if $\alpha $ equals

If the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{k}{x^{2}}\right)^{10}$ is $405,$ then $|k|$ equals 

If for positive integers $r > 1,n > 2$ the coefficient of the ${(3r)^{th}}$ and ${(r + 2)^{th}}$ powers of $x$ in the expansion of ${(1 + x)^{2n}}$ are equal, then

Number of integral tems in the expansion of  $\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}^{824}$  is equal to..................

If $a$ and $b$ are distinct integers, prove that $a-b$ is a factor of $a^{n}-b^{n}$, whenever $n$ is a positive integer.