Enter values to calculate
What is a Combination Calculator?
This calculator computes combinations (nCr), permutations (nPr), and factorials for probability and statistics problems.
This calculator computes combinations (nCr), permutations (nPr), and factorials for probability and statistics problems.
Key Formulas
- Combinations (nCr): n! ÷ (r! × (n-r)!) - Order doesn't matter
- Permutations (nPr): n! ÷ (n-r)! - Order matters
- Factorial (n!): n × (n-1) × (n-2) × ... × 1
Example: For 5 people choosing 3 for a committee: 5C3 = 10 ways. For arranging those 3 in specific roles: 5P3 = 60 ways.
Calculation Results
Enter n and r values, then select a calculation type to see results.
Calculation Details
How to Use the Combination Calculator
- Enter the total number of items (n) and the number of items to choose (r).
- Click "nCr" for combinations (order doesn't matter) or "nPr" for permutations (order matters).
- Use "n!" or "r!" buttons to calculate factorials of your input values.
- Try the preset examples to understand common applications.
This calculator handles values up to n=170 due to JavaScript's number precision limits.
Understanding Combinations vs Permutations
Combinations (nCr): Use when order doesn't matter:
- Selecting a committee from a group
- Choosing toppings for pizza
- Picking lottery numbers
- Forming teams
Permutations (nPr): Use when order matters:
- Arranging people in a line
- Creating passwords
- Race finishing positions
- Assigning specific roles
Real-World Applications
Combinatorics appears everywhere in daily life:
- Card Games: Poker hands (52C5 = 2,598,960)
- Lottery: 6 numbers from 49 (49C6 = 13,983,816)
- Genetics: DNA sequence combinations
- Computer Science: Algorithm complexity
- Quality Control: Sample testing methods
- Sports: Tournament brackets and team formations
Frequently Asked Questions
Use combinations (nCr) when the order of selection doesn't matter, like choosing 3 people from 10 for a committee. Use permutations (nPr) when order matters, like arranging 3 people in specific positions (president, vice president, secretary). If you can rearrange the selected items and get the same result, use combinations.
Factorials grow extremely quickly. 170! is near the limit of JavaScript's number precision. For larger values, the result would be infinity or lose accuracy. Most practical applications don't require factorials beyond this range. For very large combinatorial calculations, specialized mathematical software is needed.
When r (items to choose) is greater than n (total items), the result is always 0 for combinations and permutations. You cannot choose more items than you have available. For example, you can't choose 10 people from a group of 5 people.
For nCr = n! ÷ (r! × (n-r)!):
Example: 5C3 = 5! ÷ (3! × 2!) = (5×4×3×2×1) ÷ ((3×2×1) × (2×1)) = 120 ÷ (6 × 2) = 120 ÷ 12 = 10.
The calculator shows these steps to help you understand the process.
Example: 5C3 = 5! ÷ (3! × 2!) = (5×4×3×2×1) ÷ ((3×2×1) × (2×1)) = 120 ÷ (6 × 2) = 120 ÷ 12 = 10.
The calculator shows these steps to help you understand the process.