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Permutation Calculator

Calculate the number of permutations possible from a set of objects with this easy-to-use permutation calculator.

Calculate Your Permutation Calculator

Type of Permutation

Total number of items (n)

Items to select (r)

What is a Permutation?

In mathematics, a permutation refers to an arrangement of objects in a specific order. Unlike combinations where the order doesn't matter, in permutations the order is important. Permutations are used to calculate the number of ways to arrange a set of objects when the order matters.

Types of Permutations

1. Permutations without Repetition

This is the arrangement of r objects selected from a set of n distinct objects, where each object can be used only once.

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

Where n! (n factorial) is the product of all positive integers less than or equal to n.

2. Permutations with Repetition

This is the arrangement of n objects where there are repetitions (some objects are identical).

P = n! / (n₁! × n₂! × ... × nₖ!)

Where n is the total number of objects and n₁, n₂, ..., nₖ are the numbers of each type of object.

Examples of Permutations

Example 1: Arranging People in a Line

If you have 8 people and want to arrange 3 of them in a line, the number of possible arrangements is:

P(8,3) = 8! / (8-3)! = 8! / 5! = 8 × 7 × 6 = 336

Example 2: Arranging Letters with Repetition

How many different ways can the letters in the word "MISSISSIPPI" be arranged?

The word has 11 letters: 1 M, 4 I's, 4 S's, and 2 P's.

P = 11! / (1! × 4! × 4! × 2!) = 11! / (4! × 4! × 2!) = 34,650

Permutations vs. Combinations

Permutations

Order matters
Used when arranging objects in sequence
Formula: P(n,r) = n! / (n-r)!
Example: Different lock combinations

Combinations

Order doesn't matter
Used when selecting a group without regard to order
Formula: C(n,r) = n! / (r! × (n-r)!)
Example: Selecting team members

Applications of Permutations

Cryptography and password security
Genetic sequencing
Scheduling and optimization problems
Game theory and probability
Statistical analysis
Computer science algorithms

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Frequently Asked Questions

Permutations involve arranging items in a specific order, while combinations involve selecting items without regard to order. In permutations, the sequence matters (e.g., the code '123' is different from '321'). In combinations, only the selection matters, not the order (e.g., selecting cards A, B, and C is the same regardless of order).

For selecting r items from n distinct items, the formula is:

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

For permutations with repetition (when some items are identical):

P = n! / (n₁! × n₂! × ... × nₖ!)

where n is the total number of items and n₁, n₂, ..., nₖ are the counts of each type of item.

A factorial (denoted by !) is the product of a positive integer and all positive integers less than it. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials grow very quickly; 10! is already over 3.6 million, and 20! exceeds 2.4 quintillion. By definition, 0! = 1.

Yes, this is called a permutation with repetition. If you have n total items where n₁ are of one type, n₂ of another type, and so on, the formula is: n! / (n₁! × n₂! × ... × nₖ!). For example, the word 'BANANA' has 6 letters (3 A's, 1 B, 2 N's), so the number of distinct arrangements is 6! / (3!×1!×2!) = 60.

P(n,n) represents the number of ways to arrange all n items from a set of n distinct items. This is simply n! (n factorial). For example, P(5,5) = 5! = 120, meaning there are 120 different ways to arrange 5 distinct items.

Permutations have many practical applications: generating PIN codes and passwords, arranging seating at events, scheduling tasks or appointments, analyzing genetic sequences, solving cryptographic problems, and creating game strategies. They're also crucial in probability calculations for events where order matters.

Yes, order is crucial in permutations. For example, when considering permutations of the letters A, B, and C, the arrangements ABC, ACB, BAC, BCA, CAB, and CBA are all counted as different permutations because the order of elements differs in each case.

For permutations with constraints, you can often use complementary counting (find total permutations and subtract invalid ones) or break down the problem based on the constraint. For example, if you need to arrange 5 people in a row but two specific people must be adjacent, you can first treat those two people as one unit (giving you 4 units total) and multiply by the ways to arrange the two people internally (2!).

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