If coefficients of ${(2r + 1)^{th}}$ term and ${(r + 2)^{th}}$ term are equal in the expansion of ${(1 + x)^{43}},$ then the value of $r$ will be

If the coefficient of ${(2r + 4)^{th}}$ and ${(r - 2)^{th}}$ terms in the expansion of ${(1 + x)^{18}}$ are equal, then$ r=$

The term independent of $x$ in the binomial expansion of $\left( {1 - \frac{1}{x} + 3{x^5}} \right){\left( {2{x^2} - \frac{1}{x}} \right)^8}$ is

Sum of co-efficients of terms of degree $m$  in the expansion of $(1 + x)^n(1 + y)^n(1 + z)^n$ is

The sum of all those terms which are rational numbers in the expansion of $\left(2^{1 / 3}+3^{1 / 4}\right)^{12}$ is:

In the expansion of $(1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0$, the sum of the coefficient of $x^3$ and $x^{-13}$ is equal to