The analogy fails because the relationships are not congruent.
Permutations do indeed relate to ordinals, since they refer to position within an ordered set. Combinations may relate to cardinals, but cardinals would apply in the same way to permutations.
A permutation is a set/collection where order is relevant. A combination is a set/collection where order is not relevant.
A cardinal refers to a quantity, such as 'three apples'. An ordinal refers to position, such as 'the second apple'.