Question

The number of terms free from radical sign in the expansion of $(1+\sqrt[3]{33}+\sqrt[7]{77}{)}^{10}$ is

• A. $1$
• B. $6$
• C. $11$
• D. None of the above.

Answer 1

The correct option is B $6$

In the expansion of ${(1+3^{1/3}+7^{1/7})}^{10}$, the general term$=\frac{10!}{r_1!r_2!r_3!}{3}^{\frac{r_2}{3}}{7}^{\frac{r_3}{7}}$

Now, for terms free from radical sign,

$r_2$ must be a multiple of $3$ and $r_3$ must be a multiple of $7$.

Now, $r_1+r_2+r_3=10$

So, possible values:$0371000730307460190$

Hence, for the above $6$ combinations, there can be $6$ such terms.

So, $B$ is correct.