Question

Number of irrational terms in the binomial expansion of $(3^{1/5}+7^{1/3}{)}^{100}$ is

• A. $90$
• B. $88$
• C. $93$
• D. $95$
• E. $94$

Answer 1

The correct option is E $94$

The general term in the expansion of $(3^{1/5}+7^{1/3}{)}^{100}$ is
$T_{r+1}={}^{100}{C}_r\cdot 3^{(100-r)/5}\cdot 7^{r/3}$

For the term to be rational, $(100-r)$ should be divisible by $5$ and $r$ should be divisible by $3$
$\Rightarrow r$ should be divisible by $15$
$\Rightarrow r=0,15,30,45,60,75,90$
For all other values of $r$, terms will be irrational .
Thus, there are $7$ rational terms.
$\because$ The total number of terms in the given binomial expression is $101$
$\therefore$The number of irrational terms $=101-7=94$

Note: Although the correct answer is $94$, it has not been provided in any of the options of the official paper.