Question

The sum of rational terms in ${(\sqrt{2}+\sqrt[3]{33}+\sqrt[6]{65})}^{10}$ is

• A. $11792$
• B. $1260$
• C. $1264$
• D. none of these

Answer 1

The correct option is A $11792$

${(\sqrt{2}+\sqrt[3]{33}+\sqrt[6]{65})}^{10}$
General term of given expansion $=\frac{10!}{\alpha !\beta !\gamma !}{2}^{\alpha /2}{3}^{\beta /3}{5}^{\gamma /6}$
where $0\le \alpha ,\beta ,\gamma \le 10$ and $\alpha +\beta +\gamma =10$
$\Rightarrow \alpha =0,2,4,6,8,10;\beta =0,3,6;\gamma =0,6$
Hence, possible pairs of $(\alpha ,\beta ,\gamma )$ are $(4,6,0),(4,0,6),(10,0,0)$
So, there are three rational terms.
Sum of rational terms $=\frac{10!}{4!6!0!}{2}^2{3}^2{5}^0+\frac{10!}{4!0!6!}{2}^2{3}^0{5}^1+\frac{10!}{10!0!0!}{2}^5{3}^0{5}^0$
$=11792$