Combination Calculator
When Order Doesn't Matter
A combination counts how many distinct groups of r items can be pulled from a set of n, treating every rearrangement of the same group as identical. A hand of five cards is the same hand no matter what order the cards were dealt in — that's the defining feature that separates a combination from a permutation, and it's why the combination count is always smaller.
The Formula
This is the permutation formula divided by r!, which removes the duplicate counting caused by the r! different orders each group of r items could have been arranged in. As with permutations, n and r must be non-negative whole numbers with r ≤ n.
Where Combinations Apply
- Card games and lotteries — counting the number of possible hands or number combinations.
- Statistics — the binomial coefficient in probability distributions is exactly nCr.
- Team and committee selection — choosing a group where each member's role is identical, so order is irrelevant.
- Quality control sampling — counting how many ways a sample of a given size can be drawn from a batch.
| n | r | nCr |
|---|---|---|
| 10 | 3 | 120 |
| 6 | 3 | 20 |
| 52 | 5 | 2,598,960 |
52C5 = 2,598,960 is the total number of distinct 5-card poker hands possible from a standard deck.
How to Use This Calculator
- Enter the Total Items (n).
- Enter the Items Chosen (r).
- Select Calculate to see the number of possible groups.
Related Calculations
When the arrangement order does matter, switch to the Permutation Calculator. Both build on the Factorial Calculator.