The Leibniz formula expresses $\det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i=1}^{n} a_{i,\sigma(i)}$ as a sum over all permutations. For a $3 \times 3$ matrix where exactly two non-zero entries appear in each row, how many permutation terms in the Leibniz expansion are guaranteed to be non-zero?