The number of irrational terms in the expansion of
Question:

The number of irrational terms in the expansion of $\left(4^{1 / 5}+7^{1 / 10}\right)^{45}$ is

(a) 40

(b) 5

(c) 41

(d) none of these

Solution:

(c) 41

The general term $T_{r+1}$ in the given expansion is given by

${ }^{45} C_{r}\left(4^{1 / 5}\right)^{45-r}\left(7^{1 / 10}\right)^{r}$

For $T_{r+1}$ to be an integer, we must have $\frac{r}{5}$ and $\frac{r}{10}$ as integers i.e. $0 \leq r \leq 45$

$\therefore r=0,10,20,30$ and 40

Hence, there are 5 rational and 41 , i.e., $46-5$, irrational terms.