Vitamin K is needed for the liver to finish making four specific clotting factors. Without it, these factors are produced but stay inactive because they cannot undergo gamma-carboxylation, the step that gives them calcium-binding ability and lets them sit on phospholipid membranes during coagulation.
The four factors to remember are $II$, $VII$, $IX$ and $X$. The classic mnemonic is the year 1972, read as factors $10, 9, 7, 2$. The natural anticoagulants protein C and protein S share this vitamin K dependence, which is why warfarin affects both clotting and anticlotting proteins.
Scanning the answer choices, only option B, listing $IX$ and $X$, gives a pair where both members belong to the vitamin K dependent group. Every other option is spoiled by at least one factor that is not vitamin K dependent: factor $III$ (tissue factor), factor $V$, factor $VIII$ and factor $XII$ all sit outside the group.
\[\boxed{\text{IX and X}}\]