Permutations Calculator

Calculate ordered permutations P(n, r), circular arrays, repeating subsets, and multiset word anagram frequencies in high precision.

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Total Items (n)

Arrange Items (r)

Arranging 3 items in strict sequential order.

Total Unique Ordered Arrangements

720

There are exactly 720 unique arrangements possible.

📂 Mathematical Substitution Steps

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

Substitute: P(10, 3) = 10! / (10 - 3)! = 10! / 7!

Numerator (10!): 3628800

Denominator ((7)!): 5040

Result: P(10, 3) = 720

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Combinations vs Permutations

The fundamental core difference between Combinations and Permutations boils down to order.

In Permutations, the relative sequence or sequencing order matters (such as credentials passcode sequences). In Combinations, order is completely irrelevant, and only the unique elements of the chosen subset matter (like picking cards in a hand).

Combinations (nCr)

{A, B} equals {B, A}

Permutations (nPr)

(A, B) is distinct from (B, A)

Common FAQs

What does standard nPr signify?

nPr stands for the number of ordered arrangements possible when selecting 'r' items from a pool of 'n' total options.

When should I use Multiset Permutations?

Multiset formulas are applied when the source set contains repeating elements (like the word 'APPLE' containing two 'P's) to avoid double-counting identical arrangements.

What are circular permutations?

Circular permutations arrange items in a continuous loop. Since rotating the loop doesn't change relative neighbors, there is no designated 'first' position, making the formula (n - 1)!.

Counting Arrangements, Not Just Selections

The permutation situations

Order matters — that single fact multiplies the counts enormously:

Scenario	Count	Formula
Arrange all 5 books on a shelf	120	5! (full factorial)
Top 3 finishers from 12 runners	1,320	P(12,3) = 12×11×10
4-character password, no repeats, 26 letters	358,800	P(26,4)
Seating 8 guests at 8 chairs	40,320	8!
Anagram counts of a word	n! ÷ repeats!	MISSISSIPPI: 34,650

Factorials explode — that's the lesson

10! is 3.6 million; 20! exceeds 2 quintillion. This explosion is why brute-forcing arrangements fails fast, why scheduling problems are computationally hard, and why password length beats complexity (every added position multiplies the search space by the alphabet size — the math behind the password generator's length advice). It's also why exam questions love permutations: the formulas are simple, the numbers are dramatic.

The order-doesn't-matter sibling

If your problem says “choose” rather than “arrange”, divide out the orderings — that's combinations. Chance questions built on either count resolve in the probability calculator.