Question

Find the number of integral terms in the expansion of $(\sqrt{3}+\sqrt[8]{85}{)}^{256}$

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Answer 1

$T_{r+1}={}^{256}{C}_r(\sqrt{3}{)}^{256-r}{5}^{\frac{r}{8}}$
=${}^{256}{C}_r(3{)}^{\frac{256-r}{2}}{5}^{\frac{r}{8}}$
For this term to be an integer, both $\frac{256-r}{2}and\frac{r}{8}i.e(128-\frac{r}{2}and\frac{r}{8})$ should be integers. r should be a multiple of 8 for $\frac{r}{8}$ to be an integer. When r is a multiple of 8, 128 - $\frac{r}{2}$ (r<257) will also be an integer.
$\Rightarrow$ r can take the values 0,8,16.....256
$\Rightarrow$ or $0,1\times 8,2\times 8,3\times \mathrm{8.......32}\times 8$
$\Rightarrow$ Number of values r can take = 32+1= 33