Permutations Calculator (nPr)
Calculate P(n,r), the number of ordered arrangements of r items chosen from n, with the n!/(n−r)! formula, factorial cancellation, and step-by-step work shown.
How to use this calculator
Enter n (the total number of distinct items available) and r (how many you are arranging). The calculator returns P(n, r) — the number of ordered arrangements — along with the factorial formula, the cancelled-down multiplication chain, and a full step-by-step breakdown.
Order matters — the defining idea
A permutation counts ordered arrangements. Picture a race with 10 runners where you want to know the number of possible 1st/2nd/3rd place outcomes. Swapping who comes in 1st and who comes in 3rd produces a different result, even though the same three runners are on the podium. That sensitivity to order is exactly what measures.
Worked example — P(10,3)
The factorial ratio looks intimidating until you notice that cancels almost all of — everything from 7 down to 1 divides out, leaving only the top three factors: . That cancellation is why the calculator's expansion line shows a short product instead of two huge factorials: for any , only factors ever survive, counting down from .
Permutations vs. combinations
Both start from the same items, but they answer different questions:
Permutation nPr | Combination nCr | |
|---|---|---|
| Question | In how many orders can I arrange r items? | In how many ways can I select r items? |
| Formula | ||
| Example (10 people, r = 3) | President/VP/Secretary: 720 ways | 3-person committee: 120 ways |
| Relationship | — |
The relationship is direct: a combination is a permutation with the internal orderings of each group collapsed into one. .
The classic permutation-vs-combination mistake
The single most common error in counting problems is picking the wrong formula because the "order matters?" question was never asked. Two people, two prizes:
- First prize and second prize are different awards → order matters → use .
- Both winners get an identical gift card → order doesn't matter → use .
Same 2 people chosen from the same pool, but the count is different: counts each pair twice (once per ordering), while counts it once. Always ask whether swapping two chosen items changes the outcome before reaching for a formula.
Special cases
- P(n, 0) = 1 — there is exactly one way to arrange zero items: do nothing (the empty arrangement).
- P(n, n) = n! — when every item gets a position, the permutation formula reduces to a plain factorial, since .
- P(n, 1) = n — arranging a single item chosen from n is just picking which one.
Frequently asked questions
- What is P(10,3)?
- P(10,3) counts the ordered arrangements of 3 items chosen from 10. Using : .
- What does nPr mean?
- nPr — written — is the number of ways to arrange items chosen from a set of distinct items where the order of the arrangement matters. First place, second place, and third place are different outcomes even with the same three people.
- How is a permutation different from a combination?
- A permutation counts ordered arrangements; a combination counts unordered selections. Choosing a president, VP, and secretary from 10 people is a permutation (720 ways) because the roles differ; choosing an unordered 3-person committee from the same 10 people is a combination (120 ways) because no one has a distinct role.
- What is the most common mistake with permutations?
- Confusing permutations with combinations — using when order does not actually matter, or when it does. Ask first: does swapping two chosen items produce a different outcome? If yes, use ; if no, use .
- Why does P(n,0) always equal 1?
- Choosing 0 items has exactly one outcome — the empty arrangement — so for any . This matches the formula: .
- What is P(n,n)?
- When , every item gets placed, so . For example — the number of ways to line up all 5 items.
- How does nPr relate to nCr?
- . A permutation counts every ordering of the same items separately, while a combination treats those orderings as one outcome, so dividing by converts arrangements into selections. .