Quick Overview
- Calculates how many ways r elements can be chosen from a set of n using C(n,r) = n! / (r! × (n-r)!), where order does not matter.
- Supports values of n and r from 0 to 170; r must be less than or equal to n.
- Example: choosing 5 players from a roster of 11 gives C(11,5) = 462 unique combinations.
- Used in probability, statistics, lottery analysis, sports scheduling, and software QA testing.
- Suitable for students, data scientists, engineers, and sports analysts across the US and UK.
Combination Calculator — Get the Right Answer, Fast
Whether you're filling out an NCAA bracket, analyzing Powerball odds, or counting software test scenarios, combinations show up everywhere in science, sports, and daily life. The Combination Calculator tells you instantly how many ways r items can be selected from a group of n when order doesn't matter.
Formula: C(n,r) = n! / (r! × (n-r)!).
Enter your values and the tool handles the rest, displaying both the numerical result and the step-by-step formula.
What Is a Combination?
A combination is fundamentally different from a permutation. When you pick 5 cards from a deck, the order in which they're drawn doesn't change the hand — that's a combination. The binomial coefficient, written C(n,r) or "n choose r," counts exactly this. This calculator accepts: n (Total Elements) and r (Elements Chosen). Both must be whole numbers between 0 and 170, with r never exceeding n.
Formula and Reference Table
The combination formula grows rapidly with larger inputs. See common values below.
n | r | C(n,r) | Use Case |
32 | 2 | 496 | NFL matchups |
69 | 5 | 11 238 513 | Powerball draws |
52 | 5 | 2 598 960 | Poker hands |
Example 1: Teacher selects 3 from 12 students: C(12,3) = 220.
Example 2: Powerball white balls: C(69,5) = 11 238 513.
Real-World Examples
NCAA March Madness
6 conference teams in Sweet Sixteen; 4 reach Elite Eight: C(6,4) = 15 possible outcomes for analysts to model.
Powerball Lottery Odds
Matching 5 white balls from 69: C(69,5) = 11 238 513. Jackpot odds are roughly 1 in 292 million including the Powerball.
Software QA Testing
8 browser/OS environments, 3 tested per sprint: C(8,3) = 56 possible configurations to prioritize.
Fantasy Football
15 eligible running backs, pick 2 starters: C(15,2) = 105 starter-pair combinations to analyze.
Who Uses This Calculator?
High school and college students: Verify homework and exam answers quickly.
Math teachers: Generate worked examples on the fly during lessons.
Software QA teams: Count test case combinations before sprint planning.
Data scientists: Apply combinatorics in feature selection and sampling design.
Sports analysts: Calculate lineup combinations and bracket probabilities.
Finance professionals: Assess portfolio diversification scenario counts.
Researchers: Support experimental design in combinatorial trials.
Lottery enthusiasts: Understand the true odds behind ticket draws.
Conclusion and Next Steps
Combinations are one of the most versatile tools in mathematics. This calculator removes friction from the formula so you can focus on the problem. Explore also the Permutation Calculator, Probability Calculator, and Factorial Calculator.
Key Takeaways: C(n,r) = n! / (r! × (n-r)!). Order does not matter. Supported range: 0–170. C(n,0) = C(n,n) = 1. Large values like C(69,5) are computed instantly.