🎯
Combinations Calculator (nCr)
Calculate the number of ways to choose r items from n total items
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📚 Examples, Rules & Help
⚡Quick Examples - Try These Calculations
🔍How It Works
Understanding Combinations
Formula:
C(n,r)=
n!
r!×(n-r)!
Combinations count the number of ways to choose r items from n total items
Key property: Order doesn't matter
Example: Choosing 2 people from (Alice, Bob, Carol) gives 3 combinations: (Alice,Bob), (Alice,Carol), (Bob,Carol)
Formula Application
The combination formula uses factorials to account for all arrangements
Why divide by r!? Removes arrangements within selected group
Why divide by (n-r)!? Removes arrangements within unselected group
Special Cases
nC0 = 1: One way to choose nothing
nCn = 1: One way to choose everything
nC1 = n: n ways to choose one item
nCr = nC(n-r): Symmetry property
🌍Real-World Applications
🎲 Probability & Games
Lottery combinations, card games, dice probability
👥 Team Selection
Choosing committee members, forming groups, team assignments
🧬 Scientific Research
Experimental design, sample selection, statistical analysis
💼 Business Planning
Portfolio selection, resource allocation, project combinations
🎯 Strategy Games
Chess positions, game theory, competitive analysis
📊 Data Analysis
Feature selection, hypothesis testing, survey design
❓Frequently Asked Questions
What's the difference between combinations and permutations?
Combinations don't consider order (choosing 2 people for a team), while permutations do consider order (choosing 1st and 2nd place winners).
Why do we divide by factorials in the formula?
We divide by r! to remove the arrangements within the selected group, and by (n-r)! to remove arrangements within the unselected group.
Can r be larger than n?
No, you cannot choose more items than are available. If r > n, the result is 0 (impossible).
What does nC0 equal and why?
nC0 = 1 because there's exactly one way to choose nothing from any collection.
🎯Common Use Cases
🎯 Competition & Games
- • Lottery number combinations
- • Poker hand possibilities
- • Tournament bracket arrangements
- • Team formation scenarios
👥 Group Selection
- • Committee formation
- • Job interview candidate selection
- • Study group composition
- • Event planning teams
🧪 Scientific Applications
- • Clinical trial participant selection
- • Genetic combination analysis
- • Chemical compound possibilities
- • Statistical sampling methods
💼 Business Strategy
- • Investment portfolio combinations
- • Product feature selection
- • Market segment targeting
- • Resource allocation scenarios
💡Calculator Tips & Best Practices
💡Understanding the Difference
Remember: combinations are for 'choosing' (order doesn't matter), permutations are for 'arranging' (order matters).
📏Symmetry Property
nCr = nC(n-r). This means choosing r items is the same as choosing which (n-r) items to leave out.
⚠️Large Number Limitations
For very large numbers, factorials can exceed calculator limits. The calculator handles up to n=170 reliably.
⭐Practical Calculation Tip
For efficiency, always ensure r ≤ n/2. If r > n/2, calculate nC(n-r) instead using the symmetry property.
📚 References & Further Reading
Comprehensive mathematical reference for combinations and binomial coefficients
External Link
Educational resource with examples and practice problems
External Link
Note: These references provide additional mathematical context and verification of the formulas used in this calculator.