Question

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

• A. $40$
• B. $33$
• C. $26$
• D. $42$

Answer 1

The correct option is B $33$

​​​General term,
$T_{r+1}={}^{256}{C}_r\left(\sqrt{3}{)}^{256-r}\right(\sqrt[8]{85}{)}^r$
$={}^{256}{C}_r\left(3{)}^{(256-r)/2}\right(5{)}^{r/8}$
Now this term is an integer if $\frac{256-r}{2},\frac{r}{8}$ will be an integer, for which $r=0,8,16,\dots ,256$
For A.P., $0,8,16,24,...,256$ number of terms will be $\frac{256}{8}+1=33$
Hence, there are $33$ integral terms.