This permutation calculator computes nPr — the number of ways to arrange r items chosen from n when order matters — and puts the combination count beside it so the classic “does order count?” question answers itself.
nPr = n! ÷ (n − r)! — the Ordered Count
Fill r slots from n items: n choices, then n−1, then n−2… — the falling product. Divide a full factorial by the unused tail and the formula appears. Its unordered sibling lives on our Combination Calculator, and the factorials themselves on the Factorial Calculator. Part of our Math Calculators collection.
Worked Examples (Verified)
- P(10, 3) = 720 — gold/silver/bronze finishes among 10 racers: 10 × 9 × 8 ordered podiums (widget defaults)
- The same field’s unordered top-3 set: C(10, 3) = 120 — exactly 720 ÷ 3! = 720 ÷ 6
- 4-digit PINs: repetition allowed: 10⁴ = 10,000; without repetition (a stricter rule some systems use): P(10,4) = 5,040 — the security difference is a factor of two, not the chasm people assume
- Full arrangements: P(52, 52) = 52! ≈ 8.07 × 10⁶⁷ — why a shuffled deck has (essentially) never repeated in history
The One Question That Decides Everything
- Does swapping two chosen items create a different outcome? Podium vs committee, password vs pizza toppings, ranking vs raffle winners — yes = permutation, no = combination.
- Partial credit in textbooks aside, real problems mix both: choose the team by combination, then assign positions by permutation — multiply the two counts.
- Repetition flips the model entirely (n^r), and “arrangements of a multiset” (MISSISSIPPI) needs the factorial-division rule covered via our factorial page’s FAQ family.
Related Tools
- Combination Calculator — when order doesn’t matter
- Probability Calculator — these counts become probabilities
- Math Calculators — the full collection
Frequently Asked Questions
What is the permutation formula?
nPr = n! / (n − r)! — equivalently the product n × (n−1) × … × (n−r+1). Arrange all n items (r = n) and it reduces to n!, which is why 52! counts deck orderings.
Permutation vs combination — how do I choose?
Ask whether order changes the outcome. Medals, sequences, codes and rankings: permutation. Groups, hands of cards (as sets), and selections: combination. This calculator prints both so the contrast is one glance away.
How many permutations of 10 items taken 3 at a time?
720 — computed as 10 × 9 × 8. The matching combination count is 120; the ratio is 3! = 6, the arrangements of any chosen trio.
What about permutations with repetition?
n^r — each slot refills. This calculator’s repetition mode handles it: 10⁴ = 10,000 four-digit PINs versus 5,040 without repeats. Lottery draws and dice sequences split on exactly this toggle.
Why do the numbers explode so fast?
Factorial growth outruns exponentials: 20! ≈ 2.4 × 10¹⁸ already exceeds the seconds since the Big Bang. Large cases display here in scientific notation rather than pretending to print quadrillions of digits.
Is this permutation calculator free?
Yes — free, instant, no sign-up.