The number of rational terms in the binomial expansion of $(\sqrt[4]{5}+\sqrt[5]{4})^{100}$ is

We have, $$ \begin{aligned} (\sqrt[4]{5}+\sqrt[5]{4})^{100} & =\sum_{r=0}^{100}{ }^{100} C_r(\sqrt[4]{5})^{100-r}(\sqrt[5]{4})^r \\ & =\sum_{r=0}^{100}{ }^{100} C_r 5 \frac{100-r}{4} 4^{\frac{r}{5}}=\sum_{r=0}^{100} T_{r+1} \\ T_{r+1} & ={ }^{100} C_r 5 \frac{100-r}{4} 4^{\frac{r}{5}} \end{aligned} $$ Where, $\quad T_{r+1}={ }^{100} C_r 5 \frac{100-r}{4} 4^{\frac{r}{5}}$ Clearly, $T_{r+1}$ will be an integer if $\frac{100-r}{4}$ and $\frac{r}{5}$ are integers. This is possible when $100-r$ is a multiple of 4 and $r$ is a multiple of 5 $\Rightarrow 100-r=0,4,8,12, \ldots, 96,100$ and $$ r=0,5,10, \ldots, 100 $$ $\Rightarrow r=0,4,8,12, \ldots, 100$ and $r=0,5,10,100$ $\Rightarrow r=0,20,40,60,80,100$ Hence, there are 6 rational terms.