2024 AMC 10B Problem 19

Any two lattice points give a rational slope. So a line with irrational slope holds at most one lattice point: it can have $0$ (say $y = \sqrt2\,x + \tfrac12$) or exactly $1$ (say $y = \sqrt2\,x$), never two. A line with rational slope (zero included) through a lattice point $(x_0, y_0)$ also passes through $(x_0 + q, y_0 + p)$ for its reduced slope $\tfrac{p}{q},$ so it hits infinitely many; such a line has either $0$ lattice points (shift it by an irrational intercept) or more than two, never exactly one or two. So each row gives exactly two “Possible” entries. For zero and nonzero rational slope those are the “zero” and “more than two” columns; for irrational slope, the “zero” and “exactly one” columns. That’s $6$ in all. Thus, C is the correct answer.