The term independent of $x$ in the expansion of ${\left( {2x + \frac{1}{{3x}}} \right)^6}$ is

In the expansion of $(1 + x)^{43}$ if the co-efficients of the $(2r + 1)^{th}$ and the $(r + 2)^{th}$ terms are equal, the value of $r$ is :

The middle term in the expansion of ${\left( {1 - \frac{1}{x}} \right)^n}\left( {1 - {x}} \right)^n$ in powers of $x$ is

Find the $r^{\text {th }}$ term from the end in the expansion of $(x+a)^{n}$

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

If $\frac{{{T_2}}}{{{T_3}}}$ in the expansion of ${(a + b)^n}$ and $\frac{{{T_3}}}{{{T_4}}}$ in the expansion of ${(a + b)^{n + 3}}$ are equal, then $n=$