Coordinate avoidance from a nonzero permanent

If an $n$ by $n$ matrix $A$ over a field has nonzero permanent, $b\in F^n$, and every $S_i\subseteq F$ has two elements, then some $x\in {\prod }_i{S}_i$ makes every coordinate of $Ax-b$ nonzero. Apply the Combinatorial Nullstellensatz towhose coefficient of $x_1\cdots x_n$ is $\mathrm{perm}A$.