Number of Subsets Calculator

How it is calculated

For a set with n elements the tool gives four counts. Total subsets: 2n, because each element is either in a subset or out of it. Proper subsets: 2n − 1, which leaves out only the full set itself. Non-empty subsets: also 2n − 1, leaving out only the empty set. Two-element subsets: C(n, 2) = n(n − 1) ÷ 2. Here proper means every subset except the set itself, so the empty set is counted, and the proper and non-empty figures are equal. The input must be a whole number from 0 to 60; a fraction, a negative number or a larger value gives no result. All counts are exact integers, so 60 elements print every digit of 2⁶⁰ = 1,152,921,504,606,846,976, with no rounding.

Common mistakes

• Counting the empty set and the full set out of habit. The total of 2ⁿ includes both, so 5 elements give 32, not 30.
• Expecting the proper-subset figure to differ from the non-empty one. Here they are always equal, because each leaves out exactly one subset.
• Confusing subsets of any size with subsets of a fixed size. 32 counts every size, while only 10 of them have exactly two elements.
• Entering the number of subsets instead of the number of elements. For a set of 3 elements enter 3, not 8.
• Entering a set with repeated items without removing duplicates. A set counts distinct elements only, so {a, a, b} has 2 elements.