Gaussian elimination is taught with fractions and exact arithmetic. Real implementations swap rows to put the largest entry on the pivot first. What does that partial pivoting protect against?

B. Dividing by a very small pivot, which magnifies rounding error

Dividing by a tiny pivot amplifies whatever error is already there, so a mathematically irrelevant choice becomes numerically decisive. It is a clean example of the gap between an algorithm and its implementation, the algebra is indifferent to row order and the floating-point arithmetic is not.