A factorial (n!) is the product of all positive integers from 1 to n. 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1. Factorials grow extremely fast: 10! = 3,628,800; 20! is about 2.4 quintillion. They appear throughout combinatorics and probability.

MathApril 12, 2026

Permutation Calculator Guide: Permutation Formula Explained (2026)

Q: What is the permutation formula (nPr)?

P(n, r) = n! / (n − r)!, where n is total items and r is the number being arranged. For arranging 3 books from a shelf of 8: P(8, 3) = 8! / 5! = 8 × 7 × 6 = 336 different arrangements.

Q: What is a permutation with repetition?

When items can be repeated, the formula is n^r — n choices for each of r positions. A 4-digit PIN using digits 0–9: 10⁴ = 10,000 possible PINs. A 3-letter code using 26 letters with repetition: 26³ = 17,576 possible codes.

Q: How many ways can you arrange letters in a word with repeated letters?

Use the formula n! / (n₁! × n₂! × ...) where n is total letters and n₁, n₂, etc. are the frequencies of each repeated letter. For MISSISSIPPI: 11! / (1! × 4! × 4! × 2!) = 34,650 distinct arrangements.

Q: What is a circular permutation?

Circular permutations arrange items in a circle where rotations are considered identical. The formula is (n − 1)!. Seating 6 people around a round table: (6 − 1)! = 5! = 120 arrangements. This is fewer than linear arrangements (720) because rotating everyone one seat clockwise doesn't create a 'new' arrangement.

By The hakaru Team·Last updated March 2026

Quick Answer

*nPr formula: P(n,r) = n! / (n−r)! — order matters.
*With repetition: nʳ — n choices for each of r positions.
*Repeated items: n! / (n₁! × n₂! × ...) for words with duplicate letters.
*Circular: (n−1)! arrangements around a ring or table.

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What Is a Permutation?

A permutation is an arrangement of objects where order matters. The arrangements ABC, ACB, BAC, BCA, CAB, CBA are 6 different permutations of three letters. Compare this to combinations where all these would count as a single selection {A, B, C}.

The nPr Formula

P(n, r) = n! / (n − r)!

This calculates the number of ways to arrange r items from a set of n items.

Worked Example

How many 3-letter arrangements from 5 letters {A, B, C, D, E}?

P(5, 3) = 5! / (5−3)! = 5! / 2! = 120 / 2 = 60
Intuition: 5 choices for 1st position, 4 for 2nd, 3 for 3rd = 5 × 4 × 3 = 60

Factorials

n! (n factorial) = the product of all positive integers from 1 to n. Some values:

n	n!
0	1 (by convention)
5	120
10	3,628,800
15	1,307,674,368,000
20	2,432,902,008,176,640,000

Factorials grow absurdly fast. A standard deck of 52 cards can be arranged in 52! ways — a number with 68 digits, larger than the estimated atoms in the observable universe.

Permutations with Repetition

When items can be reused, each position has n choices independently: nʳ

4-digit PIN (0–9): 10&sup4; = 10,000
6-character password (26 lowercase letters): 26⁶ = 308,915,776
License plate (3 letters + 4 digits): 26³ × 10&sup4; = 175,760,000

Permutations with Identical Items

When some items are identical: n! / (n₁! × n₂! × ...)

Arrangements of MISSISSIPPI (11 letters: M×1, I×4, S×4, P×2):

11! / (1! × 4! × 4! × 2!) = 39,916,800 / 1,152 = 34,650

Circular Permutations

Around a circle, rotations are identical, so: (n − 1)!

Seating 8 people around a round table: 7! = 5,040 (vs. 40,320 in a line).

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Frequently Asked Questions

What is the permutation formula (nPr)?

P(n,r) = n! / (n−r)!. Arranging 3 from 8: P(8,3) = 8 × 7 × 6 = 336.

What is a factorial?

n! = product of all integers from 1 to n. 5! = 120. 0! = 1 by convention. Factorials grow extremely fast.

What is a permutation with repetition?

When items can repeat: nʳ. A 4-digit PIN = 10&sup4; = 10,000 possibilities.

How many ways can you arrange letters in a word with repeated letters?

n! / (n₁! × n₂! × ...). MISSISSIPPI = 11! / (4! × 4! × 2!) = 34,650.

What is a circular permutation?

(n−1)! arrangements in a circle. 6 people around a table: 5! = 120. Rotations don't count as new arrangements.