Statement – 1 : The number of integral terms in ${(\sqrt{5}+\sqrt[8]{7})}^{1024}$ is 129

Statement – 2 : The sum of rational terms in ${(\sqrt{2}+\sqrt[3]{3}+\sqrt[6]{5})}^{10}$ is 1262

Statement – 3 : The number of irrational terms in the expansion of ${(5^{1/6}+2^{1/8})}^{100}$ is 97

The correct statements among the following are

Statement – 1 : General term = ${ }^{1024}{C}_r{\left(5^{1/2}\right)}^{1024-r}{\left({\left(7\right)}^{1/8}\right)}^r$ will be integer. 

if r is multiple of 8 $\Rightarrow r=0,8,16,24,\mathrm{............},1024$ $\Rightarrow$Number of integral terms = 129.

Statement – 2 : General term $=\frac{10!}{\alpha !\beta !\gamma !}{2}^{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{3}^{\raisebox{1ex}{$\beta $}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{5}^{\raisebox{1ex}{$T$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}$ for rational terms $\alpha =0,2,4,6,8,10\beta =0,3,6\gamma =0,6$

Hence possible sets $=(4,6,0),(4,0,6);(10,0,0)$

Hence, there are s rotational terms

$\therefore$ required rdom $=\frac{10!}{4!6!}{2}^{2}5\frac{10!}{10!}{2}^5=12632$

Statement – 3 : $t_{r+1},$ the (r + 1) in the expansion of ${(5^{1/6}+2^{1/8})}^{10}$ is $t_{r+1}{=}^{100}{C}_r{\left(5^{1/6}\right)}^{100-r}{\left(2^{1/8}\right)}^r$

As 5 and 2 are relatively prime, $t_{r+1}$ will be rational if $\frac{100-r}{6}$ and $\frac{r}{8}$ are both integers. i.e if 100 – r is a multiple of 6 and r is a multiple of 8. As $0\le r\le 100$, multiple of 8 upto 100 and corresponding value of 100- r, r = 0, 8, 16, 24, ……….., 88, 96
100 – r = 100, 92, 84, 76,…….., 12, 4
Out of 100 – r, multiple of 6 are 84, 60, 36, 12
$\therefore$ there are just four rational terms
$\Rightarrow$ number of irrational terms is 101-4=97