This is probably just another way to present Fedor Petrov's solution: Expand
$$\frac{(1-t \alpha_1) (1-t \alpha_2) \cdots (1-t \alpha_{n+k-1})}{(1-t \beta_1)(1 - t \beta_2) \cdots (1-t \beta_n)}$$
as a formal power series in $t$. The coefficient of $t^k$ is a degree $k$ polynomial in the $\beta$'s, which vanishes whenever $\{ \beta_1, \beta_2, \ldots, \beta_n \}$ is a $n$-element subset of the $\alpha$'s.