Question

The sum of rational term in $(\sqrt{2}+\sqrt[3]{33}+\sqrt[5]{55}{)}^{10}$ is equal to

• A. $12632$
• B. $1260$
• C. $126$
• D. $83217$

Answer 1

The correct option is D $83217$

$S={(2^{1/2}+3^{1/3}+5^{1/5})}^{10}$

By multinomial expansion,

$\Rightarrow S=\sum \frac{\left(10\right)!}{\left(r_1\right)!\left(r_2\right)!\left(r_3\right)!}{\left(2^{1/2}\right)}^{r_1}{\left(3^{1/3}\right)}^{r_2}{\left(5^{1/5}\right)}^{r_3}$

${\Rightarrow r}_1+r_2+r_3=10$

$\therefore r_1=2{\lambda }_1$

$r_2={3\lambda }_2$ $\Rrightarrow$ For any term to be rational.

$r_3=5{\lambda }_3$

Case $1\to {\lambda }_1={\lambda }_2=0;{\lambda }_3=2$

Case $2\to {\lambda }_2={\lambda }_3=0;{\lambda }_1=5$

Case $3\to {\lambda }_1={\lambda }_2={\lambda }_3=1$

Case $4\to {\lambda }_1={\lambda }_2=2;{\lambda }_3=0$

$\therefore$ Sum of coefficient $=\frac{10!}{4!6!}\times 2^2.3^2+\frac{10!}{10!}\times 5^2+\frac{10!}{10!}\times 2^5+\frac{10!}{2!3!5!}\times 2\times 3\times 5$

$=83217.$