If the third term in the binomial expansion of $(1 + x^{\log_2 x})^5$ equals $2560$,then a possible value of $x$ is

Let $\alpha$ be the constant term in the binomial expansion of $\left(\sqrt{x} - \frac{6}{x^{3/2}}\right)^n$,$n \leq 15$. If the sum of the coefficients of the remaining terms in the expansion is $649$ and the coefficient of $x^{-n}$ is $\lambda \alpha$,then $\lambda$ is equal to $..........$.

The value of $x$ in the expression $(x + x^{\log_{10} x})^5$,if the third term in the expansion is $1,000,000$.

For a positive integer $n$,$(1+\frac{1}{x})^n$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are in the ratio $2:5:12$,then $n$ is equal to

The coefficient of $t^{50}$ in $(1 + t^2)^{25}(1 + t^{25})(1 + t^{40})(1 + t^{45})(1 + t^{47})$ is -

The sum of all possible values of $n \in N$ such that the coefficients of $x$, $x^2$, and $x^3$ in the expansion of $(1+x^2)^2(1+x)^n$ are in arithmetic progression is: