Question

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

• A. 94
• B. 88
• C. 93
• D. 95

Answer 1

Answer: (A)

General term of ${\left(3^{1/5}+7^{1/3}\right)}^{100}$ is given by
$T_{r+1}={}^{100}{C}_r{\left(3^{1/5}\right)}^{100-r}{\left(7^{1/3}\right)}^r$
$={}^{100}{C}_r\cdot 3\frac{100-r}{5}\cdot 7^{\frac{r}{3}}$
For a rational term, $\frac{100-r}{5}$ and $\frac{I}{3}$ must be integer.
Hence, $r=0,15,30,45,60,75,90$
$\therefore$ There are seven rational terms.
Hence, number of irrational terms
$=101-7=94$