, we get: Such a injective mapping from to integers is arbitrary, so this representation of all the subsets of is not unique, but the sort order of the enumerated set does not change its cardinality. (E.g., can be used to construct another injective mapping from to the integers without changing the number of one-to-one correspondences.) However, such finite binary representation is only possible if S can be enumerated. (In this example, , , and are enumerated with 1, 2, and 3 respectively as the position of binary digit sequences.) The enumeration is possible even if has an infinite cardinality (i.e., the number of elements in is infinite), such as the set of integers or rationals, but not possible for example if S is the set of real numbers, in which case we cannot enumerate all irrational numbers. Relation to binomial theorem The binomial theorem is closely related to the power set. A –elements combination from some set is another name for a –elements subset, so the number of combinations, denoted as (also called binomial coefficient) is a number of subsets with elements in a set with elements; in other words it's the number of sets with elements which are elements of the power set of a set with elements. For example, the power set of a set with three elements, has: C(3, 0) = 1 subset with 0 elements (the empty subset), C(3, 1) = 3 subsets with 1 element (the singleton subsets), C(3, 2) = 3 subsets with 2 elements (the complements of the singleton subsets), C(3, 3) = 1 subset with 3 elements (the original set itself). Using this relationship, we can compute using the formula: Therefore, one can deduce the following identity, assuming : Recursive definition If is a finite set, then a recursive definition of proceeds as follows: If , then . Otherwise, let and ; then . In words: The power set of the empty set is a singleton whose only element is the empty set. For a non-empty set , let be any element of the set and its relative complement; then the power set of is a union of a power set of and a power set of whose each element is expanded with the element. Subsets of limited cardinality The set of subsets of of cardinality less than or equal to is sometimes denoted by or , and the set of