Combinations Calculator (nCr)
Calculate the number of ways to choose r items from n total items
📚 Examples, Rules & Help
⚡Quick Examples of Combinations
📐Combinations Formula
Where n = total number of items, r = number of items to choose, and ! represents factorial
🔍How to Calculate Combinations
How to Calculate
Step 1: Identify n (total items) and r (items to choose)
Step 2: Calculate n! (n factorial)
Step 3: Calculate r! (r factorial)
Step 4: Calculate (n-r)! ((n-r) factorial)
Step 5: Apply formula: C(n,r) = n! ÷ (r! × (n-r)!)
Example: Choose 3 items from 5 total (C(5,3))
• n = 5, r = 3
• n! = 5! = 5 × 4 × 3 × 2 × 1 = 120
• r! = 3! = 3 × 2 × 1 = 6
• (n-r)! = (5-3)! = 2! = 2 × 1 = 2
• C(5,3) = 120 ÷ (6 × 2) = 120 ÷ 12 = 10
Understanding Combinations
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
❓Frequently Asked Questions
What's the difference between combinations and permutations?
Why do we divide by factorials in the formula?
Can r be larger than n?
What does nC0 equal and why?
🎯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