Combination Calculator
Combination Calculator
Use combinations when you are selecting a group and the order does not matter.
Enter your values
Enter values and select Calculate.
Combination Calculator – Calculate nCr Online
Use this free Combination Calculator to calculate the number of ways to choose a group of items when order does not matter. Enter the total number of available items (n) and the number you want to select (r) to calculate nCr, also called “n choose r”. This combinations calculator is useful for mathematics homework, probability, statistics, team selection and other counting problems.
Combination formula (nCr)
Here, n is the total number of items, r is the number selected, and ! represents factorial. The valid range is 0 ≤ r ≤ n.
The formula divides the number of ordered arrangements by the number of ways each selected group can be rearranged. This removes duplicate arrangements and leaves each unique group counted once.
How to use the Combination Calculator
- Enter the total number of available items in Total items (n).
- Enter how many items you want to choose in Items chosen (r).
- Select Calculate to see the number of possible combinations.
- Use Clear to reset both inputs and start another calculation.
Worked combination examples
5C2 = 5! ÷ (2! × 3!) = 10. There are 10 different pairs.
8C3 = 8! ÷ (3! × 5!) = 56. There are 56 possible groups of three.
6C1 = 6. Each of the six items forms one possible selection.
4C4 = 1. There is only one way to select every item.
Combinations vs permutations
Use combinations when the selected group is what matters and changing the order does not create a new outcome. Use permutations when arrangement or position matters. For example, selecting three people for a committee is a combination; assigning first, second and third places is a permutation.
Important properties of combinations
- nC0 = 1: there is one way to select nothing.
- nCn = 1: there is one way to select all items.
- nCr = nC(n−r): choosing r items to include is equivalent to choosing n−r items to leave out.
- 0 ≤ r ≤ n: the selection count cannot be negative or exceed the total.
Where are combinations used?
Combinations are used in probability calculations, lottery selections, committee formation, card games, sampling, quality control and statistical analysis whenever the order of selected items is irrelevant.
Frequently asked questions about combinations
What is a combination?
A combination is a selection of items where the order does not matter. For example, choosing three team members is the same group regardless of selection order.
What is the combination formula?
The formula is nCr = n! ÷ [r! × (n − r)!], where n is the total number of items and r is the number selected.
How do I use this combination calculator?
Enter the total number of items (n), enter the number to choose (r), and select Calculate. The result gives the number of distinct groups.
What is the difference between permutations and combinations?
Combinations ignore order, while permutations count different arrangements as distinct outcomes.
Can r be greater than n?
No. The number selected cannot exceed the total number available, so r must be between 0 and n.
What is n choose 0?
There is exactly one way to choose no items from a set, so nC0 = 1.
What is n choose n?
There is exactly one way to choose all available items, so nCn = 1.
Why does the combination formula use factorials?
Factorials count arrangements and the denominator removes the repeated orderings of the same selected group.
Are combinations used in probability?
Yes. Combinations help count possible groups and outcomes when order is irrelevant, including many probability and lottery problems.
Can combinations be used for large numbers?
Yes, but the result grows quickly. Large inputs may produce values beyond the exact integer range supported by ordinary browser number calculations.