Combinatorics Calculator

Calculate permutations, combinations, and other combinatorial values with this easy-to-use tool. Perfect for probability problems, statistics, and discrete mathematics.

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)
Select whether you want to calculate combinations, permutations, or both
Choose whether repetition is allowed in your selection
Click Calculate to see your results

Formula Used

Combination (nCr) = n! / (r! × (n-r)!)

Permutation (nPr) = n! / (n-r)!

Combination with Repetition = (n+r-1)! / (r! × (n-1)!)

Permutation with Repetition = n^r

Where:

n = Total number of items
r = Number of items to select or arrange
! = Factorial (n! = n × (n-1) × ... × 2 × 1)

Example Calculation

Real-World Scenario:

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

Given:

Total Items (n) = 10
Items to Select (r) = 3
Calculation Type = Combination
Repetition Allowed = No

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 lottery odds and probabilities
Determining password security and possibilities
Analyzing genetic combinations and inheritance
Optimizing resource allocation and scheduling

Key Benefits

Quickly solve complex counting problems
Understand probability in real-world scenarios
Make informed decisions based on possible outcomes
Save time on manual calculations in statistics

Common Mistakes & Tips

Remember that combinations are for selection where order doesn't matter, while permutations are for arrangements where order does matter. For example, selecting a committee is a combination problem, while arranging books on a shelf is a permutation problem.

Be careful to use the correct formula when repetition is allowed. With repetition, combinations use the formula (n+r-1)!/(r!×(n-1)!) while permutations use n^r. Forgetting to account for repetition can lead to significantly incorrect results.

Remember that 0! = 1 by definition. Also, be careful with large factorials as they grow very quickly. When working with combinations and permutations, you can often simplify calculations by canceling terms before multiplying everything out.

Frequently Asked Questions

Combinations are used when the order of selection doesn't matter (e.g., selecting a committee), while permutations are used when the order does matter (e.g., arranging books on a shelf). For the same values of n and r, there will always be more permutations than combinations.

Use the "allow repetition" option when items can be selected more than once. For example, when selecting letters for a password, you can use the same letter multiple times. When dealing with physical objects like cards from a deck, repetition is typically not allowed.

Combinations have fewer results because they treat different orderings of the same items as identical. For example, selecting {A, B, C} is the same as selecting {C, B, A} in combinations, but they are considered different arrangements in permutations. This is why we divide by r! in the combination formula.

References & Disclaimer

Mathematical Disclaimer

This calculator provides combinatorial calculations for educational and informational purposes. While we strive for accuracy, results should be verified for critical applications. The calculator may have limitations with very large numbers due to computational constraints.

References

Combination - Wikipedia - Mathematical definition and properties of combinations
Permutation - Wikipedia - Mathematical definition and properties of permutations
Combinations and Permutations - Math is Fun - Interactive explanation with examples

Accuracy Notice

This calculator uses JavaScript's built-in Number type, which has precision limitations for very large factorials. For extremely large values of n and r, results may be approximated using scientific notation. For precise calculations with large numbers, specialized mathematical software is recommended.

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

Combinatorics Calculator

Results