This is probably just another way to present Fedor Petrov's solution: Expand $$\frac{(1-t \alpha_1) (1-t \alpha_2) \cdots (1-t \alpha_{n-k+1})}{(1-t \beta_1)(1 - t \beta_2) \cdots (1-t \beta_k)}$$$$\frac{(1-t \alpha_1) (1-t \alpha_2) \cdots (1-t \alpha_{n+k-1})}{(1-t \beta_1)(1 - t \beta_2) \cdots (1-t \beta_n)}$$ as a formal power series in $t$. The coefficient of $t^n$$t^k$ is a degree $n$$k$ polynomial in the $\beta$'s, which vanishes whenever $\{ \beta_1, \beta_2, \ldots, \beta_k \}$$\{ \beta_1, \beta_2, \ldots, \beta_n \}$ is a $k$$n$-element subset of the $\alpha$'s.
Adjusted roles of n and k to match the question. Also fixed typo in original subscripts on alpha's. (At least I think I have it right here!)
Louis Deaett
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