Permutation & Combination Calculator

Permutation & Combination Calculator

Count arrangements and selections of distinct items without replacement, using exact whole-number arithmetic.

Quick Answer

Choose 3 from 5: 5C3 = 10 groups, while 5P3 = 60 ordered arrangements. The difference is whether order matters.

nPr = n!/(n − r)!   |   nCr = n!/[r!(n − r)!]

Use whole numbers with 0 ≤ r ≤ n ≤ 500. Items must be distinct and selected without replacement.

Combination nCr
10

5C3 = 10 selections; 5P3 = 60 ordered arrangements.

Combinations nCr10
Permutations nPr60
Orderings per group (r!)6
n − r2
Steps

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    How to Use the Permutation & Combination Calculator

    Enter the number of distinct available items n and how many you select r. Choose whether the main result should be nCr or nPr. Both exact counts appear in the result panel. This tool assumes no item is selected twice.

    Permutation Versus Combination

    Use a permutation for ordered roles, such as gold, silver, and bronze places. Use a combination for an unordered group, such as selecting three people for a team. Each group of r distinct items can be arranged r! ways, so nPr = nCr × r!.

    nPr and nCr Formulas

    For 0 ≤ r ≤ n, permutations multiply n by the next r − 1 descending whole numbers. Combinations divide that ordered count by r! because order does not create a new group. Zero selected items produce one empty selection and one empty arrangement.

    nPr = n!/(n − r)!
    nCr = n!/[r!(n − r)!]

    Worked Example: Choose Three From Five

    For five distinct items, ordered triples number 5 × 4 × 3 = 60. Each unordered triple has 3! = 6 arrangements, so 5C3 = 60/6 = 10. If positions are different, use 60; if a selected group counts once, use 10.

    When These Formulas Do Not Apply

    The standard nPr and nCr formulas assume distinct items and selection without replacement. If repeats are allowed or items are indistinguishable, another counting model is needed. The tool rejects r greater than n and does not silently switch models.

    Permutation & Combination FAQs

    What is 5C3?

    5C3 = 10.

    What is 5P3?

    5P3 = 60.

    Why is nPr larger than nCr?

    An unordered group can be arranged in r! orders, so nPr = nCr × r!.

    Editorial review

    SolverCalc names Prof. Dr. Khalil Mudassar as its educational reviewer. The formulas and examples on this page follow the references below.

    Sources

    View references

    Disclaimer

    For educational use. Check your class or textbook convention when the problem defines a different range, sequence index, or notation.