Combination Calculator
Calculate the number of ways to select items from a set
This calculator helps you find the number of ways to select items from a set when the order doesn't matter:
- Enter the total number of items (n) in your set
- Enter the number of items you want to choose (r)
- The calculator will:
- Calculate the number of possible combinations
- Show the formula used
- Break down the calculation steps
- Display the final result
A combination represents the number of ways to select items from a set when the order doesn't matter:
- The formula for combinations is:
C(n,r) = n! / (r! × (n-r)!)
- Key properties:
- Order doesn't matter (unlike permutations)
- You can't choose more items than you have (r ≤ n)
- C(n,r) = C(n,n-r) (symmetry property)
- C(n,0) = C(n,n) = 1
Combinations are used in many real-world applications:
- Probability calculations
- Team selection and group formation
- Game theory and strategy
- Statistical sampling
- Lottery and gambling odds
- Molecular biology (DNA combinations)
- Computer science (algorithm analysis)
What's the difference between combinations and permutations?
Combinations don't consider the order of selection, while permutations do. For example, selecting cards A and B is the same combination as B and A, but they are different permutations.
Why do we use factorials in the combination formula?
Factorials help us count all possible arrangements and then divide out the arrangements that we consider the same (when order doesn't matter). This gives us the correct count of unique combinations.
Is there a limit to the numbers I can input?
Yes, due to computational limitations, very large numbers may cause overflow errors. For practical purposes, we recommend keeping n below 100. For larger numbers, consider using approximation methods or logarithmic calculations.