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The one-parameter family of polynomials $J_n(x,y)={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{y+j}{j}\right)x^j$ is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each $n\ge 6$, the polynomial $J_n(x,y_0)$ is irreducible over $\mathbb{Q}$ for all but finitely many $y_0\in \mathbb{Q}$. If $n$ is odd, then with the exception of a finite set of $y_0$, the Galois group of $J_n(x,y_0)$ is $S_n$; if $n$ is even, then the exceptional set is thin.