The factorial of a cardinal, denoted by, is the cardinality of the set of all permutations of a set which is of cardinality. We give a condition that makes the cardinal equality provable without the axiom of choice. In fact, we prove in that, for all cardinals, if and there is a permutation without fixed points on a set which is of cardinality, then.

The factorial of a cardinal, denoted by, is the cardinality of the set of all permutations of a set which is of cardinality. We give a condition that makes the cardinal equality provable without the axiom of choice. In fact, we prove in that, for all cardinals, if and there is a permutation without fixed points on a set which is of cardinality, then.