The constant term in the expansion of $\left(x^{2}-\frac{1}{x^{2}}\right)^{16}$ is

Arrange the expansion of $\left(x^{1/2} + \frac{1}{2x^{1/4}}\right)^n$ in decreasing powers of $x$. Suppose the coefficients of the first three terms form an arithmetic progression. Then,the number of terms in the expansion having integer power of $x$ is

For $n \in N$,in the expansion of $\left(\sqrt[4]{x^{-3}}+a \sqrt[4]{x^5}\right)^n$,the sum of all binomial coefficients lies between $200$ and $400$ and the term independent of $x$ is $448$. Then the value of $a$ is

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

If the $7^{th}$ term in the binomial expansion of $\left( \frac{3}{\sqrt[3]{84}} + \sqrt{3} \ln x \right)^9, x > 0,$ is equal to $729,$ then $x$ can be

If $(1-x+x^2)^{10}=a_0+a_1 x+a_2 x^2+\ldots+a_{20} x^{20}$,then $2 a_2+3 a_3+4 a_4+\ldots+20 a_{20}=$