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 |
If you like this calculator
Please help us simply by sharing it. It will help us a lot!
Related Calculators
Other calculators you might find useful.
Sector Area Calculator
Calculate the area, radius, or central angle of a circular sector
Normal Distribution Calculator
Calculate z-scores, probabilities, and values for normal distributions
Square Root Calculator
Calculate principal and secondary square roots, and identify perfect squares.
Chi-Square Calculator
Calculate chi-square statistics, p-values, and test for goodness of fit or independence between categorical variables
Radical Simplifier Calculator
Simplify square roots instantly to their exact simplest radical form.
Derivative Calculator
Calculate derivatives and find the slope of functions at any point
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)