What Is a Permutation & Combination Calculator?
A permutation and combination calculator computes the number of ways to arrange or select items from a set. Permutations count ordered arrangements (where sequence matters), while combinations count unordered selections. These concepts are foundational in probability, statistics, and combinatorics.
How to Use This Calculator
- Enter the total number of items (n).
- Enter the number of items to choose (r).
- Click Calculate to see both the permutation P(n,r) and combination C(n,r) results along with the formulas and factorial values used.
Key Concepts
Permutations: P(n, r) = n! ⁄ (n − r)! counts ordered arrangements. Combinations: C(n, r) = n! ⁄ (r! × (n − r)!) counts unordered selections. The factorial function n! = n × (n−1) × ... × 2 × 1 grows extremely fast: 10! = 3,628,800 and 20! exceeds 2.4 × 1018. Combinations with repetition use the formula C(n+r−1, r). Pascal's triangle provides a visual representation where each entry is C(n, r).
Frequently Asked Questions
When do I use permutations vs. combinations?
Use permutations when the order of selection matters (e.g., arranging books on a shelf, assigning rankings). Use combinations when order does not matter (e.g., choosing a committee, selecting lottery numbers).
What is a factorial and why does it grow so fast?
A factorial (n!) multiplies all positive integers from 1 to n. It grows faster than exponential functions because each subsequent term is larger. By Stirling's approximation, n! ≈ √(2πn) × (n⁄e)n.
What are real-world applications of combinations?
Lottery odds use combinations: choosing 6 from 49 gives C(49, 6) = 13,983,816 possibilities. Card hand probabilities, committee selection, and sampling methods all rely on combinatorial calculations.