If the number of integral terms in the expansion of ${\left(3^{1/2}+5^{1/8}\right)}^{\mathrm{n}}$ is exactly 33 then the least value of n is

In ${\left(3^{1/2}+5^{1/8}\right)}^{\mathrm{n}};{\mathrm{T}}_{\mathrm{r}+1}{=}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}{\left(3^{1/2}\right)}^{\mathrm{n}-r}{\left(5^{1/8}\right)}^{\mathrm{r}}$

If ${\mathrm{T}}_{\mathrm{r}+1}$ is an integer then r is a multiple of 8 and n - r is a multiple of 2.

Number of integral terms $=33\Rightarrow 33$ multiple in $0,8,16,24,\dots \text{ is }256\Rightarrow \mathrm{n}=256$.