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

The coefficient of $x^{24}$ in the expansion of $(1+x^2)^{12}(1+x^{12})(1+x^{24})$ is

If the coefficient of $x^3$ in the binomial expansion of $x^3(2 \sqrt{3} x^2 + \frac{1}{kx})^{12}$ is $880$,then $k$ is equal to

The number of rational terms in the binomial expansion of $(4^{1/4} + 5^{1/6})^{120}$ is $....$

If the $r$-th and $(r+1)$-th terms in the expansion of $(p+q)^{n}$ are equal,then the value of $\frac{(n+1)q}{r(p+q)}$ is

If the $k^{\text{th}}$ term in the expansion of $\left(\frac{3}{2} x^2 - \frac{1}{3x}\right)^6$ is independent of $x$,then the numerically greatest term in the expansion of $\left(\frac{3}{2} x^2 - \frac{1}{3x}\right)^k$ when $x = \frac{2}{3}$ is: