2. How many rational terms are there in the expansion of the binomial $(\sqrt{3}+\sqrt[3]{2})^{100}$?
Solution. The general term in the binomial expansion is $T_{k+1}=\binom{100}{k} 3^{\frac{100-k}{2}} 2^{\frac{k}{3}}$, $k \in\{0,1, \ldots, 100\}$. For the term to be rational, $\frac{100-k}{2}$ and $\frac{k}{3}$ must be integers. Since $\frac{100-k}{2}$ is an integer if 2 is a divisor of $k$, and $\frac{k}{3}$ is an integer if 3 is a divisor of $k$, we get that $k$ must be an integer divisible by 6. Therefore, the number of rational terms is the number of integers divisible by 6 from 0 to 100, which is $\left[\frac{100}{6}\right]+1=17$.