Combination Calculator
Calculate the number of ways to choose r items from n items where order does not matter. Uses the formula C(n,r) = n!/(r!(n-r)!)
This free online combination calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
Total number of items in the set.
Number of items to choose from the set.
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Combination Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Combination Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
Formula Reference
Combination Calculator Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
When to Use This Calculator
- •Use the Combination Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
About This Calculator
The Combination Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the number of ways to choose r items from n items where order does not matter. Uses the formula C(n,r) = n!/(r!(n-r)!) The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.
About Combination Calculator
The combination calculator computes the number of ways to select r items from a set of n items when the order of selection does not matter. This is one of the most fundamental concepts in combinatorics and probability theory. Combinations appear everywhere: lottery odds, committee formation, poker hand rankings, genetic trait combinations, and sampling plans in quality control. The key distinction from permutations is that in combinations, {A, B, C} and {C, B, A} count as the same selection. The binomial coefficient, often written as 'n choose r', is also the coefficient of x^r in the expansion of (1+x)^n, which is why it appears throughout algebra, calculus, and discrete mathematics.
The Math Behind It
Formula Reference
Combination Formula
C(n, r) = n! / (r! * (n - r)!)
Variables: n = total number of items; r = number of items chosen; ! = factorial
Worked Examples
Example 1: Lottery odds
A lottery requires choosing 6 numbers from 49. How many possible combinations are there?
There are 13,983,816 possible lottery combinations, giving odds of about 1 in 14 million.
Example 2: Committee selection
How many ways can a 4-person committee be formed from a group of 12 people?
There are 495 possible committees.
Common Mistakes & Tips
- !Confusing combinations with permutations. If order matters, use the permutation formula instead.
- !Forgetting that 0! = 1, which means C(n, 0) = C(n, n) = 1.
- !Using combinations when items can be repeated; for that, use the multiset coefficient C(n+r-1, r).
Related Concepts
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Frequently Asked Questions
When should I use combinations instead of permutations?
Use combinations when the order does not matter (e.g., choosing team members). Use permutations when the order matters (e.g., assigning ranked positions).
What is the largest value of n this calculator can handle?
JavaScript can handle factorials accurately up to about n = 170 before exceeding the maximum floating-point value. For larger values, logarithmic computation or big-integer libraries are needed.
What is Pascal's triangle?
Pascal's triangle is an arrangement of binomial coefficients where each entry equals the sum of the two entries directly above it. Row n contains C(n,0), C(n,1), ..., C(n,n).