Question

The number of integral terms in the expansion of $(5^{1/2}+7^{1/6}{)}^{642}$ is

• A. $108$
• B. $106$
• C. $103$
• D. $109$

Answer 1

Answer: (A)

$108$

Expansion of ${(5^{1/2}+7^{1/6})}^{642}$

General term $=T_{r+1}={{}^{642}C}_r{\left(5^{1/2}\right)}^{642-r}{\left(7^{1/6}\right)}^r$

$={{}^{642}C}_r{5}^{\frac{642-r}{2}}{7}^{r/6}$

For term to be integral $\frac{r}{6}$ should be integer

$\Rightarrow$ $r=0,6,12,18,...642$ at these values

$\frac{642-r}{2}$ is also integral

$\Rightarrow$ Number of integral terms are $=108$