The total number or irrational terms in the binomial expansion of $\left( {{7^{1/5}} - {3^{1/10}}} \right)^{60}$ is 

In the expansion of ${({5^{1/2}} + {7^{1/8}})^{1024}}$, the number of integral terms is

If the coefficients of $(r-5)^{th}$ and $(2 r-1)^{th}$ terms in the expansion of $(1+x)^{34}$ are equal, find $r$

Let the coefficients of $x ^{-1}$ and $x ^{-3}$ in the expansion of $\left(2 x^{\frac{1}{5}}-\frac{1}{x^{\frac{1}{5}}}\right)^{15}, x>0$, be $m$ and $n$ respectively. If $r$ is a positive integer such $m n^{2}={ }^{15} C _{ r } .2^{ r }$, then the value of $r$ is equal to

Let $[t]$ denotes the greatest integer $\leq t$. If the constant term in the expansion of $\left(3 x^2-\frac{1}{2 x^5}\right)^7$ is $\alpha$, then $[\alpha]$ is equal to $............$.

The coefficient of ${x^{39}}$ in the expansion of ${\left( {{x^4} - \frac{1}{{{x^3}}}} \right)^{15}}$ is