Permutations vs. Combinations β When Order Matters and When It Doesn't
Permutations and combinations are the two fundamental tools for counting the number of ways to select or arrange items from a set. The key distinction: if the order of selection matters, use permutations; if only the selection itself matters (not the order), use combinations.
Formulas and real-world examples:
Permutation: nPr = n! / (nβr)! β Ordered selection of r from n
Combination: nCr = n! / (r! Γ (nβr)!) β Unordered selection of r from n
Relationship: nPr = nCr Γ r!
1. Lottery β Choosing 6 numbers from 45 (order doesn't matter): C(45,6) = 8,145,060 ways
2. PIN codes β Choosing 4 digits from 10 without repetition (order matters): P(10,4) = 5,040 ways
3. Team selection β Picking 3 members from 10 for a committee: C(10,3) = 120 ways
4. Race results β Predicting 1st, 2nd, and 3rd place from 8 runners: P(8,3) = 336 ways
5. Card hands β The number of possible 5-card poker hands from 52: C(52,5) = 2,598,960
6. Probability β P(event) = favorable outcomes / total outcomes, both calculated via combinations
Factorials grow extremely fast β 20! exceeds 2.4 Γ 10ΒΉβΈ. For very large n, the calculator may display results in exponential notation when the value exceeds JavaScript's safe integer limit.
Frequently Asked Questions
A: By convention, 0! = 1. There is exactly one way to arrange zero objects (doing nothing). This definition ensures nC0 = nCn = 1 and makes the combination formula work for edge cases.
A: Yes. nC0 = 1 because there is exactly one way to choose nothing from n items. Similarly, nCn = 1 because there is exactly one way to choose all n items.
A: No, this calculator handles only standard permutations and combinations without repetition. With-repetition permutations = nΚ³; with-repetition combinations = C(n+rβ1, r).