The Permutation and Combination Calculator finds how many ways r items can be chosen from a set of n, when order matters (permutations) and when it does not (combinations).
| Permutations nPr | 30 |
| Combinations nCr | 15 |
| Permutations with repetition nʳ | 36 |
| Combinations with repetition | 21 |
Calculation steps:
| nPr = n! ÷ (n − r)! | 6! ÷ 4! = 30 |
| nCr = n! ÷ (r!(n − r)!) | 6! ÷ (2! × 4!) = 15 |
A permutation is an arrangement where order matters. The number of ways to arrange r items chosen from n is nPr = n! ÷ (n − r)!. Example: picking a president and vice-president from 6 people gives 6 × 5 = 30 permutations.
A combination is a selection where order does not matter. nCr = n! ÷ (r!(n − r)!). Choosing 2 committee members from 6 gives 30 ÷ 2 = 15 combinations, because each pair was counted twice as a permutation.
If items can be reused, there are nʳ ordered arrangements (like a 4-digit PIN: 10⁴ = 10,000) and (n + r − 1)! ÷ (r!(n − 1)!) unordered selections (like choosing scoops of ice cream from n flavours).
Lottery odds (choosing 6 numbers from 49: 13,983,816 combinations), passwords, seating plans, tournament schedules and card hands all use these formulas.
What is n!? n factorial, the product 1 × 2 × … × n; 0! = 1.
Why is nPr ≥ nCr? Each combination can be ordered r! ways.
How do I remember the difference? Permutation = position matters.