Permutation and Combination Calculator

Calculate permutations and combinations for a given set of items and selections. Find the number of possible arrangements or selections with or without repetition.

How to Use This Calculator

Enter the total number of items in your set (n)
Enter the number of items you want to select or arrange (r)
Choose between permutation (order matters) or combination (order doesn't matter)
Select whether to include repetition in your calculation
Click Calculate to see the number of possible arrangements or selections

Formula Used


            Permutation without repetition: P(n,r) = n! / (n-r)!

            Permutation with repetition: P(n,r) = n^r

            Combination without repetition: C(n,r) = n! / (r! × (n-r)!)

            Combination with repetition: C(n,r) = (n+r-1)! / (r! × (n-1)!)

Where:

n = Total number of items in the set
r = Number of items to select or arrange
! = Factorial (n! = n × (n-1) × ... × 1)

Example Calculation

Real-World Scenario:

A committee of 3 people needs to be formed from a group of 10 candidates. How many different committees can be formed?

Given:

Total number of items (n) = 10 candidates
Number of items to select (r) = 3 committee members
Calculation type = Combination (order doesn't matter)
Repetition = No (a person can't be selected twice)

Calculation:

C(10,3) = 10! / (3! × (10-3)!) = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120

Result: 120 different committees can be formed

Why This Calculation Matters

Practical Applications

Calculating password possibilities for security analysis
Determining seating arrangements for events
Analyzing genetic combinations in biology
Optimizing inventory and supply chain decisions

Key Benefits

Helps understand probability and likelihood of events
Essential for statistical analysis and research
Optimizes decision-making in various fields
Foundation for more advanced mathematical concepts

Common Mistakes & Tips

Remember that permutations consider order (ABC is different from BAC), while combinations don't (ABC is the same as BAC). If you're selecting items where the order doesn't matter (like a committee), use combinations. If you're arranging items where order matters (like a password), use permutations.

When repetition is allowed, you can select the same item multiple times. This significantly changes the calculation. For example, when selecting 3 digits from 0-9 with repetition allowed, you have 10³ = 1000 possibilities (000 to 999). Without repetition, you have P(10,3) = 10×9×8 = 720 possibilities.

When working with factorials, remember that 0! = 1 by definition. Also, be careful with large factorials as they grow extremely quickly. For combinations, you can simplify calculations by canceling terms before multiplying to avoid working with unnecessarily large numbers.

Frequently Asked Questions

Permutations consider the order of selection, while combinations do not. For example, if selecting 2 letters from A, B, C, permutations would count AB and BA as different, while combinations would count them as the same selection.

Use calculations with repetition when items can be selected more than once. Examples include passwords where digits can repeat, selecting multiple items of the same type from inventory, or genetic inheritance where the same allele can be passed down.

By definition, 0! = 1. This convention makes many mathematical formulas work consistently. For example, C(n,0) = n!/(0!×n!) = 1, which makes sense as there's exactly one way to select zero items from any set.

References & Disclaimer

Mathematical Disclaimer

This calculator provides mathematical calculations based on the inputs provided. Results are accurate according to combinatorial mathematics principles, but real-world applications may have additional constraints not accounted for in this calculator.

References

Combinations and Permutations - Math is Fun - Basic explanation of permutations and combinations with examples
Permutation - Wikipedia - Comprehensive mathematical definition and properties
Combination - Wikipedia - Detailed explanation of combinations in mathematics

Accuracy Notice

This calculator provides results for standard permutation and combination calculations. For very large numbers (typically > 10^70), results may be displayed in scientific notation. For applications requiring precise calculations with extremely large numbers, specialized mathematical software may be necessary.

About the Author

Kumaravel Madhavan

Web developer and data researcher creating accurate, easy-to-use calculators across health, finance, education, and construction and more. Works with subject-matter experts to ensure formulas meet trusted standards like WHO, NIH, and ISO.

Connect with LinkedIn

Related Calculators

Permutation and Combination Calculator

Results