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Permutation Calculator

Choose statistical parameters, enter their values in the designated fields, and use this calculator to determine permutations of order in possible subset notation.

Names:

Types to choose from?

(n)

Number Chosen?

(r)

Result:

Related

An online permutation calculator helps you determine the number of possible arrangements (permutations) of a set of items, where the order of elements matters. This nPr calculator finds the number of permutations when choosing r objects from a total of n items in a dataset.

Keep reading to learn about the permutation formula, step-by-step manual calculations, and how our permutation calculator simplifies this process.

Permutation Calculator

You can also try our online combination calculator to quickly calculate combinations for any dataset.

Permutation Formula

The number of permutations of n objects taken r at a time is:

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

Where:

n = Total number of items
r = Number of items selected
P(n,r) = Total permutations

Permutation with Repetition

If some elements are identical, the formula becomes:

P(n,r) = n! n₁! n₂! ... nₖ!

n₁, n₂, ..., nₖ = Number of identical elements of each type
! = Factorial of a number (product of all positive integers up to that number)

To quickly calculate factorials, you can use our online factorial calculator.

Step-by-Step Permutation Calculation

Example 1:

Suppose a company's name starts with three letters. If available letters are S, P, D, F, I, J, how many three-letter arrangements are possible if each letter is used only once?

Solution:

Total letters: n = 6
Letters chosen: r = 3

P(6,3) = 6! / (6-3)! = 6! / 3!

Step 1: Expand factorials:

P(6,3) = (6 × 5 × 4 × 3!) / 3!

Step 2: Cancel out 3!:

P(6,3) = 6 × 5 × 4 = 120

Answer: There are 120 possible permutations of three letters.

Example 2:

At the start of a race with 15 competitors, calculate the number of permutations for the top three medalists.

Solution:

Total competitors: n = 15
Medal positions: r = 3

P(15,3) = 15! / 12!

Step 1: Expand and cancel factorials:

P(15,3) = 15 × 14 × 13 = 2730

Answer: There are 2,730 possible permutations for the top three medalists.

Permutations vs Combinations

Permutations consider order of selection, while combinations do not. You can calculate combinations from permutations:

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

Using the nPr (Permutation) Calculator

Steps to use our calculator:

Select the type of elements (numbers, letters, etc.)
Enter total elements (n)
Enter elements to choose (r)
Choose permutations or permutations with repetition
Click "Calculate"

Output:

Total permutations
Permutations with repetition (if selected)
Step-by-step calculation details

FAQs

What is permutation used for?

Permutations are used in:

Computer Science: Sorting algorithms, data arrangements
Physics: Particle state arrangements
Biology: RNA/DNA sequence analysis

How do you do permutations?

In permutations, order matters. For example, a locker code 4587 cannot be opened with 8574.

Conclusion

Permutation calculations are essential in mathematics, science, and everyday problem-solving. Using an online permutation calculator provides fast, accurate results for both small and large datasets.

References

Other languages: Turkish, Polish, Indonesian, German, Japanese, Korean, and more.

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