Combination Calculator
Calculate combinations C(n,r) with step-by-step solutions and real-world examples
C(n, r) = n! / (r! × (n-r)!)
Also known as "n choose r" - the number of ways to choose r items from n items (order doesn't matter)
Must be a non-negative integer
Must satisfy 0 ≤ r ≤ n
Quick Examples
Invalid Input
C(, ) = " choose "
ways to choose items from items
Combination (Order doesn't matter)
C(, )
n! / (r! × (n-r)!)
Permutation (Order matters)
P(, )
n! / (n-r)!
P(, ) = C(, ) × !
= ×
All Combinations for n =
Sum of all values = 2 =
! (n factorial)
! (r factorial)
! ((n-r) factorial)
Step-by-Step Solution
Identify values
n = (total items), r = (items to choose)
Apply the formula
C(, ) = ! / (! × !)
Calculate factorials
= / ( × )
Final result
C(, ) =
Symmetry Property
By symmetry: C(, ) = C(, )
Choosing items
Leaving out items
Common Combinations
| C(n,r) | Value | Real-World Example |
|---|---|---|
| C(6, 2) | 15 | Handshakes among 6 people |
| C(10, 3) | 120 | 3-person committees from 10 |
| C(52, 5) | 2,598,960 | 5-card poker hands |
| C(49, 6) | 13,983,816 | Lottery 6/49 combinations |
| C(12, 5) | 792 | Picking 5 starters from 12 players |
| C(20, 3) | 1,140 | Choosing 3 toppings from 20 |
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About Combination Calculator
What is a Combination?
A combination is a selection of items from a larger set where order does not matter. Unlike permutations, combinations only count unique selections regardless of arrangement.
Combination Formula
The number of combinations is calculated using:
C(n, r) = n! / (r! × (n-r)!)
Where:
- n = Total number of items in the set
- r = Number of items being chosen
- ! = Factorial (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120)
Combination vs Permutation
| Aspect | Combination | Permutation |
|---|---|---|
| Order | Does NOT matter | Matters |
| Formula | C(n,r) = n!/(r!(n-r)!) | P(n,r) = n!/(n-r)! |
| Example | {A,B,C} = {C,B,A} | ABC ≠ CBA |
| Use case | Selecting teams | Ranking contestants |
Real-World Examples
Lottery
Choosing 6 numbers from 49: C(49,6) = 13,983,816 possible combinations
Poker
5-card hands from 52 cards: C(52,5) = 2,598,960 combinations
Committee Selection
Choosing 3 members from 10: C(10,3) = 120 ways
Team Formation
Selecting 5 starters from 12 players: C(12,5) = 792 ways
Key Properties
Symmetry
C(n, r) = C(n, n-r)
Choosing r items is equivalent to choosing which (n-r) items to leave out.
Sum Rule
For any n: C(n,0) + C(n,1) + ... + C(n,n) = 2^n
Pascal's Identity
C(n, r) = C(n-1, r-1) + C(n-1, r)
Common Combination Values
| C(n,r) | Value | Context |
|---|---|---|
| C(5,2) | 10 | Handshakes among 5 people |
| C(10,3) | 120 | 3-person committees from 10 |
| C(52,5) | 2,598,960 | Poker hands |
| C(49,6) | 13,983,816 | Lottery 6/49 |
| C(20,3) | 1,140 | Choosing 3 from 20 |
Applications
- Probability - Calculating odds in games of chance
- Statistics - Sampling and hypothesis testing
- Computer Science - Algorithm analysis and optimization
- Genetics - Gene combination possibilities
- Chemistry - Molecular arrangements
Tips for Calculations
- Use the symmetry property when r > n/2
- Cancel common factors when computing factorials
- Remember: C(n,0) = C(n,n) = 1
- C(n,1) = n (n ways to choose 1 item)