Combination Calculator
Calculate combinations (n choose k) for selecting items where order doesn't matter. Find C(n,k), nCr, and binomial coefficients with step-by-step solutions.
C(10, 3)
120
Step-by-Step Calculation
C(n, r) = n! / (r! × (n-r)!)
C(10, 3) = 10! / (3! × 7!)
= 3,628,800 / (6 × 5,040)
= 120
10! (n factorial)
3,628,800
3! (r factorial)
6
7! ((n-r) factorial)
5,040
P(10, 3) - Permutation
720
Interpretation
There are 120 different ways to choose 3 items from a set of 10 items when the order of selection doesn't matter.
Note: If order mattered (permutation), there would be 720 ways —3! = 6 times more.
Common Examples
- • Lottery: Choosing 6 numbers from 49 = C(49,6) = 13,983,816
- • Poker hand: 5 cards from 52 = C(52,5) = 2,598,960
- • Committee: 3 people from 10 = C(10,3) = 120
Key Properties
- • C(n, 0) = C(n, n) = 1
- • C(n, r) = C(n, n-r) (symmetry)
- • C(n, r) = C(n-1, r-1) + C(n-1, r) (Pascal's identity)
- • Order does NOT matter in combinations
About This Calculator
The Combination Calculator instantly computes the number of ways to select items from a larger set when the order of selection does not matter. Whether you are calculating lottery odds, determining how many different teams can be formed, figuring out card hand possibilities, or solving combinatorics problems in mathematics, this calculator provides accurate results with step-by-step solutions. Combinations are represented as C(n,k), nCr, or "n choose k" and are fundamental in probability, statistics, and discrete mathematics. Unlike permutations where order matters, combinations only count distinct groups - selecting items A, B, C is the same as selecting C, B, A. Our calculator handles large numbers efficiently using optimized algorithms, displays the mathematical formula and calculation steps, and provides practical examples to help you understand the concept. Perfect for students studying probability and statistics, teachers creating lesson materials, researchers calculating sample combinations, and anyone curious about the mathematics of selection and choice.
How to Use the Combination Calculator
- 1Enter n: the total number of items in your set.
- 2Enter k: the number of items you want to select.
- 3The calculator instantly shows C(n,k) = n! / (k!(n-k)!).
- 4View the step-by-step factorial calculation breakdown.
- 5See practical examples of what your combination represents.
- 6Use the results for probability calculations if needed.
- 7Compare with permutations to understand when order matters.
Combination Formula Explained
The combination formula calculates the number of ways to choose k items from n items without regard to order.
Combination Formula:
C(n,k) = n! / (k! * (n-k)!)
Where:
- n = total number of items
- k = number of items to choose
- n! = n factorial (n * (n-1) * (n-2) * ... * 1)
Example: C(5,2) = 5! / (2! * 3!) = 120 / (2 * 6) = 10
This means there are 10 ways to choose 2 items from 5 items.
Common Combination Values
Reference table for frequently used combinations:
| n | k | C(n,k) | Example |
|---|---|---|---|
| 6 | 2 | 15 | Pairs from 6 people |
| 10 | 3 | 120 | Committees of 3 from 10 |
| 52 | 5 | 2,598,960 | Poker hands |
| 49 | 6 | 13,983,816 | 6/49 lottery combinations |
| 20 | 4 | 4,845 | Teams of 4 from 20 |
| 12 | 5 | 792 | Selecting 5 from 12 |
Lottery Odds:
- Powerball (5 from 69 + 1 from 26): 1 in 292,201,338
- Mega Millions (5 from 70 + 1 from 25): 1 in 302,575,350
Combinations vs Permutations
Understanding when to use combinations versus permutations:
| Aspect | Combinations | Permutations |
|---|---|---|
| Order | Does NOT matter | Matters |
| Formula | n!/(k!(n-k)!) | n!/(n-k)! |
| Notation | C(n,k), nCk | P(n,k), nPk |
| Result | Smaller | Larger |
Use Combinations When:
- Selecting committee members
- Choosing lottery numbers
- Picking card hands
- Forming teams or groups
Use Permutations When:
- Arranging items in order
- Creating passwords
- Ranking contestants
- Seating arrangements
Pro Tips
- 💡Remember: in combinations, order does NOT matter (ABC = CBA).
- 💡C(n,k) = C(n, n-k) - choosing k items equals leaving out n-k items.
- 💡For lottery odds, use combinations since drawn numbers are not ordered.
- 💡Large factorials can cause overflow - our calculator handles this efficiently.
- 💡Pascal's Triangle row n contains all values of C(n,k) for k = 0 to n.
Frequently Asked Questions
In combinations, the order does not matter - ABC is the same as CBA. In permutations, order matters - ABC is different from CBA. For the same n and k, permutations always give a larger result because they count different arrangements separately.
