Permutation in Math: Definition, Formula & Examples

Definitionnoun

What is a permutation in math?

A permutation is an arrangement of objects in a specific order, so ABC and CBA count as different. The number of ways to arrange r objects chosen from n different objects is P(n, r) = n!(n − r)!. For example, picking a president and vice president from 5 people gives 5 × 4 = 20 permutations.

In Spanish: permutación
Also called: arrangement, ordering, nPr, P(n, r)
Grade level: Precalculus

Algebra 2PrecalculusStatisticsCCSS HSS-CP.B.9SATACTASVAB

Permutation in a picture

In a permutation the same objects in a different order make a new arrangement. Filling ordered slots one at a time gives 5 × 4 = 20, which matches P(5, 2) = 5!3!.

Formal definition and notation

A permutation of r objects from a set of n distinct objects (0 ≤ r ≤ n) is an ordered selection of r of them. Their number is P(n, r) = n(n − 1) ⋯ (n − r + 1), which equals n!(n − r)! using factorials. A permutation of all n objects is an ordering of the whole set, and there are n! of them. If some objects are identical, the count of distinct orderings is n!k₁! k₂! ⋯.

Symbol	Read it as	Meaning
P(n, r)	P of n, r; the permutations of n things taken r at a time	the number of ordered arrangements of r objects from n
nPr	n P r	the same number, as written on calculators; also printed with small n and r beside the P
n!	n factorial	n × (n − 1) × ⋯ × 1, the number of orderings of all n objects

Key facts

Order matters in a permutation. If order does not matter (a team, a committee, a hand of cards), use a combination instead.
P(n, n) = n! and P(n, 1) = n. Also P(n, 0) = 1: there is one way to arrange nothing.
The counting principle is behind the formula: n choices for the first spot, n − 1 for the second, and so on.
Permutations count is always at least the combinations count: P(n, r) = C(n, r) × r!. For 5 people choose 2: 20 = 10 × 2.
Identical objects reduce the count: the letters of BOOK can be arranged in 4!2! = 12 distinct ways, not 24.

Permutation formula

P(n, r) = n!(n − r)!

n: the number of different objects available
r: how many are chosen and placed in order
n!: n factorial

Shortcut: multiply r numbers counting down from n. P(8, 3) = 8 × 7 × 6 = 336. With repeated letters, divide by the factorial of each repeat count.

Permutation examples, step by step

Example 1: Use the formula

Warm-up

Find P(6, 2).

P(6, 2) = 6!(6 − 2)! = 6!4!.
Cancel 4!: 6 × 5 × 4!4! = 6 × 5.
6 × 5 = 30.

Answer: P(6, 2) = 30.

Example 2: Medals in a race

Core

Eight runners are in a final. In how many ways can the gold, silver and bronze medals be awarded?

Order matters: gold, silver and bronze are different prizes, so this is a permutation.
n = 8 runners, r = 3 medals.
P(8, 3) = 8 × 7 × 6.
8 × 7 × 6 = 336.

Answer: The medals can be awarded in 336 ways.

Example 3: Arrangements with repeated letters

Test-style

How many different arrangements of the letters in the word ALGEBRA are there?

ALGEBRA has 7 letters, so 7! = 5040 if every letter were different.
The letter A appears twice. Swapping the two A’s gives the same word, so each arrangement was counted 2! = 2 times.
Divide: 50402 = 2520.

Answer: There are 2,520 different arrangements.

Common mistakes

✗ MistakeUsing a permutation to count 3-person committees from 8 people: P(8, 3) = 336.

✓ InsteadA committee has no order, so use a combination: 3363! = 56.

Ask whether swapping two chosen people creates a new outcome. For medals it does (permutation); for a committee it does not (combination).

✗ MistakeP(8, 3) = 8!3! = 6720.

✓ InsteadP(8, 3) = 8!(8 − 3)! = 8!5! = 336.

The bottom of the formula is (n − r)!, the factorial of the number of objects not chosen.

✗ MistakeThe letters of BOOK can be arranged in 4! = 24 ways.

✓ Instead4!2! = 12 distinct arrangements.

The two O’s look the same, so swapping them does not make a new word. Divide by the factorial of each repeat count.

Not to be confused with

Permutationvs.Combination

A combination is a selection where order does not matter. From A, B, C, D, choosing 2 gives 6 combinations (AB = BA) but 12 permutations (AB and BA both count).

Permutationvs.Factorial

n! counts the orders of all n objects. A permutation count P(n, r) orders only r of them; P(n, n) = n! is the special case.

Permutationvs.Probability

Permutations count outcomes; probability compares counts. The chance that a random order of A, B, C is exactly ABC is 13! = 16.

Quick check: 3 questions

Question 1 of 3

In how many ways can 4 different books be arranged on a shelf?

Show answer and explanation

Answer: D. All 4 books are placed in order: 4! = 4 × 3 × 2 × 1 = 24. 4 counts only the first spot, 16 is 4², and 10 adds 4 + 3 + 2 + 1.

Question 2 of 3

Which situation is a permutation?

Show answer and explanation

Answer: C. In a PIN, 1234 and 4321 are different codes, so order matters. Toppings, a card hand and invited friends are the same group in any order, so those are combinations.

Question 3 of 3

What is P(7, 3)?

Show answer and explanation

Answer: A. P(7, 3) = 7 × 6 × 5 = 210. 35 is the combination C(7, 3), 21 is 7 × 3, and 840 is 7 × 6 × 5 × 4, one factor too many.

Permutation: frequently asked questions

What is the difference between a permutation and a combination?

In a permutation order matters; in a combination it does not. Choosing a president and a vice president from 5 people is a permutation (20 ways), because Ana-president-Ben-vice differs from the reverse. Choosing a 2-person committee from the same 5 is a combination (10 ways). Ask: does rearranging the chosen items create a new outcome?

How do you calculate a permutation?

Use P(n, r) = n!(n − r)!, where n is the number of objects and r is how many you arrange. A faster way is to multiply r numbers counting down from n: P(9, 3) = 9 × 8 × 7 = 504. Calculators usually have an nPr key that does this directly.

What if some objects are the same?

Divide by the factorial of each group of identical objects. The word LEVEL has 5 letters with two L’s and two E’s, so it has 5!2! × 2! = 1204 = 30 distinct arrangements. Without dividing, every arrangement would be counted once for each way of swapping the identical letters.

Practice and tools

Permutations and Combinations CalculatorCompute nPr and nCr with every step Factorial CalculatorEvaluate the factorials in the formula Probability CalculatorTurn a permutation count into a probability

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSS-CP.B.9

Cite this entry

Effortless Math Team. “Permutation.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/permutation/

Flashcard

TermPermutationP(n, r) = n!(n − r)!

DefinitionAn arrangement where order matters. Gold, silver and bronze among 8 runners: 8 × 7 × 6 = 336 permutations.

Study the full deck

Browse all terms in the Math Dictionary · More probability terms · Random term

NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2FactorialThe factorial of a whole number n, written n!, is the product of all the whole numbers from n down to 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. It counts the ways to arrange n different objects in a row. By agreement, 0! = 1.Probability · Algebra 2, Precalculus, StatisticsCombinationA combination is a selection of items from a group in which the order does not matter. Picking Ana, Ben and Cy for a team is the same combination as Cy, Ana and Ben. The number of ways to choose r items from n is C(n, r) = n!r!(n − r)!; for example, C(5, 2) = 10.Probability · Algebra 2, Precalculus, StatisticsOutcomeAn outcome is one possible result of a chance experiment, such as rolling a 4 on a number cube or getting tails on a coin flip. The set of all outcomes is the sample space, and a group of outcomes is an event. When outcomes are equally likely, each one has probability 1 divided by the number of outcomes.Probability · Grade 7, Grade 8, StatisticsProbabilityProbability measures how likely an event is, from 0 for impossible to 1 for certain. For equally likely outcomes it is the number of favorable outcomes divided by the total number of outcomes: for a fair coin, the probability of heads is 12. Probabilities can be written as fractions, decimals or percents.Probability · Grade 6, Grade 7, Grade 8ProductThe product is the answer you get when you multiply two or more numbers. In 4 × 6 = 24, the product is 24, and 4 and 6 are the factors. In word problems, the words “product of” always mean multiply, so “the product of 3 and 7” is 21.Operations · Grade 3, Grade 4, Grade 5Dependent EventsDependent events are two events where the outcome of the first changes the probability of the second. Drawing two cards without putting the first one back is the classic example. For dependent events, P(A and B) = P(A) × P(B ∣ A), where P(B ∣ A) is the probability of B once A has happened.Probability · Grade 7, Algebra 2, StatisticsIndependent EventsIndependent events are two events where the outcome of one does not change the probability of the other. Flipping heads on a coin does not affect rolling a 5 on a die. For independent events, multiply: P(A and B) = P(A) × P(B), so P(heads and 5) = 12 × 16 = 112.Probability · Grade 7, Algebra 1, Algebra 2Two-Way TableA two-way table shows counts for two categorical variables collected from the same group, with one variable in the rows and the other in the columns. Each inside cell is a joint frequency, and the totals in the margins are marginal frequencies. Dividing a cell by a row or column total gives relative frequencies that reveal associations.Statistics · Grade 8, Algebra 1

Permutation

What is a permutation in math?

Permutation in a picture

Formal definition and notation

Key facts

Permutation formula

Permutation examples, step by step

Example 1: Use the formula

Example 2: Medals in a race

Example 3: Arrangements with repeated letters

Common mistakes

Not to be confused with

Quick check: 3 questions

Permutation: frequently asked questions

What is the difference between a permutation and a combination?

How do you calculate a permutation?

What if some objects are the same?

Related terms

Practice and tools

Sources

Flashcard

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