Subset Calculator — Power Set & C(n, k)

Fórmula

How this is calculated

For a set with n distinct elements, each element can independently be either included or excluded from a subset. That binary choice for each of n elements gives 2 × 2 × … × 2 = 2ⁿ possible selections, including the empty set ∅ and the full set itself. This total is called the power set. Proper subsets exclude one or both endpoints depending on convention; the most common definition excludes the full set itself, giving 2ⁿ − 1 proper subsets. Non-empty subsets exclude only the empty set, also giving 2ⁿ − 1.

The number of subsets of exactly k elements is the binomial coefficient C(n, k) = n! / (k! × (n−k)!), computed here with the multiplicative formula (n × (n−1) × … × (n−k+1)) / k! to avoid large intermediate factorials. These values form the n-th row of Pascal's triangle — the bar chart shows C(n, 0), C(n, 1), …, C(n, n), which follows a bell shape peaking at the middle and sums to 2ⁿ.

The input is capped at n = 30 to keep results within JavaScript's exact 53-bit integer range (2³⁰ ≈ 10⁹). For larger n the total subset count exceeds what can be shown exactly; the C(n, k) formula still works but large values are shown in scientific notation.