If $A$ and $B$ are the coefficients of $x^n$ in the expansions of $(1 + x)^{2n}$ and $(1 + x)^{2n - 1}$ respectively,then

If the coefficients of $x^9$ and $x^{10}$ in the binomial expansion of $(3+\frac{x}{2})^n$ are equal,then $n=$

If $t_r$ is the $r^{\text{th}}$ term in the expansion of $(2^x + 4^{-x})^8$ and if $t_3 = 7t_2$,then $x =$

If the coefficients of the $(2r+1)^{\text{th}}$ term and the $(r+1)^{\text{th}}$ term in the expansion of $(1+x)^{42}$ are equal,then $r$ can be

If the coefficient of the $3^{\text{rd}}$ term from the beginning in the expansion of $\left(ax^2 - \frac{8}{bx}\right)^9$ is equal to the coefficient of the $3^{\text{rd}}$ term from the end in the expansion of $\left(ax - \frac{2}{bx^2}\right)^9$,then the relation between $a$ and $b$ 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