The middle term in the expansion of $(1 + x)^{2n}$ is

If the coefficient of $x^{8}$ in $\left(a x^{2}+\frac{1}{b x}\right)^{13}$ is equal to the coefficient of $x^{-8}$ in $\left(a x-\frac{1}{b x^{2}}\right)^{13},$ then $a$ and $b$ will satisfy the relation

If the sum of the coefficients of the first,second,and third terms of the expansion of $(x^2 + \frac{1}{x})^m$ is $46$,then the coefficient of the term that does not contain $x$ is:

The middle term in the expansion of $(1 + x)^{2n}$ is

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

If the coefficient of $x^{15}$ in the expansion of $(ax^3 + \frac{1}{bx^{1/3}})^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $(ax^{1/3} - \frac{1}{bx^3})^{15}$,where $a$ and $b$ are positive real numbers,then for each such ordered pair $(a, b):$