The middle term in the expansion of ${\left( {x + \frac{1}{x}} \right)^{10}}$ is

In the expansion of ${\left( {x - \frac{1}{x}} \right)^6}$, the constant term is

Expand using Binomial Theorem $\left(1+\frac{ x }{2}-\frac{2}{ x }\right)^{4}, x \neq 0$

If in the expansion of ${(1 + x)^m}{(1 - x)^n}$, the coefficient of $x$ and ${x^2}$ are $3$ and $-6$ respectively, then m is

If the coefficients of $a^{r-1}, a^{r}$ and $a^{r+1}$ in the expansion of $(1+a)^{n}$ are in arithmetic progression, prove that $n^{2}-n(4 r+1)+4 r^{2}-2=0$

The number of integral terms in the expansion of ${({5^{1/2}} + {7^{1/6}})^{642}}$ is