If the coefficients of ${x^2}$ and ${x^3}$ in the expansion of ${(3 + ax)^9}$ are the same, then the value of $a$ is

Let $\alpha$ be the constant term in the binomial expansion of $\left(\sqrt{ x }-\frac{6}{ x ^{\frac{3}{2}}}\right)^{ n }, n \leq 15$. 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 $..........$.

If the coefficients of $x^4, x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the maximum value of $n$ is :

If the coefficients of $x^7$ in $\left( ax ^2+\frac{1}{2 bx }\right)^{11}$ and $x ^{-7}$ in $\left(a x-\frac{1}{3 b x^2}\right)^{11}$ are equal, then

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

Number of rational terms in the expansion of ${\left( {\sqrt 2 \,\, + \,\,\sqrt[4]{3}} \right)^{100}}$ is :