P Permutation Calculator — P(n, r)

Calculate the number of ordered arrangements (permutations) of r items chosen from n items. P(n, r) = n! / (n − r)!. Enter n and r to get the result, step-by-step working, and all possible arrangements for small values.

Permutations nPr — Order Matters

BrainyCalculators editorial insight — unique to this tool

P(n,r) counts ordered arrangements — a 4-digit PIN with no repeats is P(10,4) = 5,040 possibilities, while a 4-person podium from 10 runners is P(10,3) = 720 medal orderings. Scheduling meetings in sequence and anagram counts (when letters are unique) are permutation problems. Always larger than the corresponding combination: P(n,r) = r! × C(n,r).

When to use this calculator

Use permutations when position or sequence changes the outcome. For unordered groups (teams, subsets, hands), use Combination.

Reference	Value	Context
P(10,3)	720	Podium orderings
P(10,4) PIN	5,040	No repeated digits
n!	All orderings	Full shuffle of n
P(n,n)	n!	Complete permutations

Not what you need? For choosing committees where order is irrelevant, use Combination. For coin-flip style likelihood, use Probability.

Choosing a subset without caring about order?

This page counts ordered arrangements (nPr). For committees, lottery tickets, and unordered groups, use the Combination Calculator →

n — Total Items

Number of items in the set (0–200)

r — Items to Arrange

Number of items to select and arrange

Permutations Count Ordered Arrangements

A 4-digit PIN with no repeated digits has P(10,4) = 10×9×8×7 ordered arrangements — 5,280 possibilities, not C(10,4) = 210 unordered sets. Permutations matter whenever position changes meaning: race medals, tournament brackets, password length, or assigning distinct roles to people from a pool.

This calculator handles nPr (no repetition) and explains the factorial structure step by step. Dividing P(n,r) by r! recovers C(n,r) — the same r people can be arranged r! ways, which is why committees and committees-with-chairs differ by exactly that factor.

Use combinations when the label “committee,” “subset,” or “lottery ticket” fits. Use permutations when the label “ranking,” “schedule,” or “code” fits. Probability rules on this site combine event logic; they do not replace counting how many ordered outcomes exist in the sample space.

Permutation Formula for Ordered Slots

P(n, r) = n! / (n − r)!

Example: P(12, 5) = 12 × 11 × 10 × 9 × 8

Relationship: P(n, r) = C(n, r) × r!

The denominator only removes unused candidates. It does not divide by r!, because slot order is the result: first speaker, second speaker, third speaker, and so on are distinct assignments.

A permutation problem usually contains words like ranked, scheduled, seated, coded, first-to-last, or assigned to roles. Those labels make each position meaningful.

Permutation Examples

P(n, r)	Calculation	Result	Real-world meaning
P(5, 3)	5 × 4 × 3	60	Ways to arrange 3 from 5 runners on a podium
P(4, 2)	4 × 3	12	4-digit lock: 2 different digits
P(6, 6)	6!	720	Ways to arrange 6 books on a shelf
P(10,3)	10 × 9 × 8	720	Gold, silver, bronze from 10 athletes
P(52,2)	52 × 51	2,652	Ordered pairs from a deck of cards

Worked Example: Schedule 5 Talks from 12 Proposals

1

Identify the slots

There are 5 agenda positions. A keynote in slot 1 is different from the same talk in slot 5, so order matters.
2

Start filling positions

The first slot has 12 choices, the second has 11, then 10, 9, and 8 after each selected talk is removed.
3

Multiply the ordered choices

P(12, 5) = 12 × 11 × 10 × 9 × 8 = 95,040 possible schedules.
4

Compare with combinations

The unordered set count is only 792. Multiplying by 5! creates the 120 orderings of every selected talk set.

How the Permutation Calculator Works

Formula, assumptions, and calculation steps for this statistics tool.

Formula Used

P(n, r) = n! / (n - r)!

Methodology

Statistics calculators organize sample data, apply the selected descriptive or inferential formula, and report the statistic with interpretation.

Calculation Steps

Enter raw values or summary statistics.
Clean separators and count the sample size.
Apply the relevant statistic, probability, or confidence formula.
Display the result with context such as degrees of freedom, percentile, or strength.

Assumptions and Limits

Samples should be representative of the population being studied.
Normality or independence assumptions apply only where the selected method requires them.
Rounded results may differ slightly from spreadsheet software.

Frequently Asked Questions

What is a permutation?

A permutation is an ordered arrangement of items selected from a set. The order of selection matters — {A, B} and {B, A} are counted as two different permutations. P(n, r) counts all possible ordered arrangements of r items chosen from a pool of n distinct items.

Permutation vs combination — what's the difference?

In a permutation, order matters. In a combination, it does not. P(5, 3) = 60 but C(5, 3) = 10. Every combination of r items corresponds to r! permutations, so P(n, r) = C(n, r) × r!.

When does order matter?

Order matters when the arrangement itself is significant: rankings (1st, 2nd, 3rd place), passwords, scheduling tasks in sequence, or assigning distinct roles to people. If you just need to pick a group without roles, use combinations.

Real-world permutation examples?

Medal standings in a race (gold/silver/bronze), arranging letters in a word, assigning unique ID codes, scheduling jobs on a machine, planning a playlist order, and seating people in numbered chairs all involve permutations.

Real-World Applications

🏁

Podium and Ranking Outcomes

Gold-silver-bronze from 10 finalists is P(10, 3), because the same three athletes in different medal positions represent different outcomes.

🔢

PINs Without Repeated Digits

A four-position PIN from digits 0-9 without repetition has 10 × 9 × 8 × 7 sequences. The code 1234 is not interchangeable with 4321.

🪑

Assigned Seats and Roles

Seating 5 guests in numbered chairs or assigning CEO/CFO/COO from a candidate pool requires ordered slots. Each chair or role changes the outcome identity.

📅

Conference Programme Order

Choosing talks is only the start; the timetable order affects audience flow, room turnover, and keynote placement. That schedule is a permutation of selected talks.

🧬

Sequence Motifs

Short DNA or peptide motifs are ordered strings. ACG and GCA contain the same letters but code different biological patterns, so sequence counting is permutation-style.

🚚

Route Stop Ordering

A delivery route visiting the same stops in a different sequence can change mileage and arrival windows. Ordered route plans are counted as arrangements, not sets.

Common Mistakes

Using permutations when combinations are needed

Permutations count ordered arrangements; combinations count unordered selections. Choosing a 5-person committee from 20 candidates is a combination problem (order doesn't matter). Using P(20, 5) instead of C(20, 5) overstates the count by 5! = 120 times.

Assuming repetition is not allowed when it is

Standard nPr formulas assume no repetition — each item can be selected at most once. If repetition is allowed (e.g., a 4-digit PIN where each digit can be reused), the formula is n^r (10^4 = 10,000), not P(10, 4) = 5,040.

Confusing P(n, r) with P(r, n)

P(n, r) is n!/(n−r)! — choosing r from n items. P(r, n) would be r!/(r−n)! — a different (and usually meaningless) calculation. Always ensure n ≥ r; if r > n, P(n, r) = 0 as you cannot choose more items than exist in the set.

Treating identical objects as distinct

The permutation formula counts all objects as distinguishable. If some items are identical (e.g., arranging the letters in AABB), the formula overstates the count — divide by the factorials of each group of identical items to get the correct number of distinct arrangements.

Not checking whether restrictions apply

Many real-world permutation problems have constraints — certain items must or cannot be adjacent, a specific item must be first, etc. The basic P(n, r) formula applies only to unrestricted permutations. Constrained problems require adjusted counting using the multiplication principle or complementary counting.

Permutation vs Combination Quick Reference

Feature	Permutation P(n,r)	Combination C(n,r)
Order matters?	Yes	No
Formula	n! / (n−r)!	n! / (r! × (n−r)!)
P(5,3) vs C(5,3)	60	10
P(n,r) vs C(n,r)	C(n,r) × r!	P(n,r) / r!
Example problem	Race finishing positions	Lottery number selection
Repetition variant	n^r	C(n+r−1, r)

References

Rosen, K.H. Discrete Mathematics and Its Applications. McGraw-Hill, 2018.
Feller, W. An Introduction to Probability Theory and Its Applications. Wiley, 1968.
Tucker, A. Applied Combinatorics. Wiley, 2012.
Grimaldi, R.P. Discrete and Combinatorial Mathematics. Pearson, 2003.
Graham, R., Knuth, D., and Patashnik, O. Concrete Mathematics. Addison-Wesley, 1994.