If $a^3 + b^6 = 2$, then the maximum value of the term independent of $x$ in the expansion of  $(ax^{\frac{1}{3}}+bx^{\frac{-1}{6}})^9$ is, where $(a > 0, b > 0)$

Show that the coefficient of the middle term in the expansion of $(1+x)^{2 n}$ is equal to the sum of the coefficients of two middle terms in the expansion of $(1+x)^{2 n-1}$

In the binomial expansion of ${\left( {a - b} \right)^n},n \ge 5,\;$ the sum of $5^{th}$ and $6^{th}$ terms is zero , then $a/b$ equals.

The coefficient of ${x^n}$in expansion of $(1 + x)\,{(1 - x)^n}$ is

Find an approximation of $(0.99)^{5}$ using the first three terms of its expansion.

If the fourth term in the expansion of $\left(x+x^{\log _{2} x}\right)^{7}$ is $4480,$ then the value of $x$ where $x \in N$ is equal to