Editing Power set (section)

== Relation to binomial theorem ==
The [[binomial theorem]] is closely related to the power set. A {{math|''k''}}–elements combination from some set is another name for a {{math|''k''}}–elements subset, so the number of [[combination]]s, denoted as {{math|C(''n'', ''k'')}} (also called [[binomial coefficient]]) is a number of subsets with {{mvar|k}} elements in a set with {{mvar|n}} elements; in other words it's the number of sets with {{math|k}} elements which are elements of the power set of a set with {{math|n}} elements.

For example, the power set of a set with three elements, has:
* {{math|1=C(3, 0) = 1}} subset with {{math|0}} elements (the empty subset),
* {{math|1=C(3, 1) = 3}} subsets with {{math|1}} element (the singleton subsets),
* {{math|1=C(3, 2) = 3}} subsets with {{math|2}} elements (the complements of the singleton subsets),
* {{math|1=C(3, 3) = 1}} subset with {{math|3}} elements (the original set itself).

Using this relationship, we can compute {{math|{{abs|2<sup>''S''</sup>}}}} using the formula:
<math display="block">\left|2^S \right | = \sum_{k=0}^{|S|} \binom{|S|}{k} </math>

Therefore, one can deduce the following identity, assuming {{math|1={{abs|''S''}} = ''n''}}:
<math display="block">\left |2^S \right| = 2^n = \sum_{k=0}^{n} \binom{n}{k} </math>