Combination Calculator
Count combinations when order doesn’t matter, such as committee picks, lottery draws or product bundles. Compare with permutations, and see the working and the actual list for small cases.
Calculate
Choose r from n without repeats, order ignored. nCr = n! ÷ (r! × (n − r)!)
Result
The formula
Standard combinations choose r items from n without order: nCr = n! ÷ (r! × (n − r)!). With repetition allowed, combinations are C(n + r − 1, r). Permutations count ordered selections: nPr = n! ÷ (n − r)!.
Worked example
- Standard combination with n = 5 and r = 3.
- nCr = 5! ÷ (3! × 2!) = (5 × 4 × 3) ÷ (3 × 2 × 1).
- Result: 10 unique groups.
Result: 5C3 = 10
How combination counts are computed
Combination formulas count unordered groups. Selecting the same members in a different order is still one combination.
Standard combination (nCr)
Use this when you select r distinct items from n distinct options and order does not matter. Valid for non-negative integers with n ≥ r.
Combination with repetition
Use this when items can repeat in a group (for example, choosing scoops where flavours can repeat). Formula: C(n + r − 1, r).
Relation to permutation
Permutations count arrangements; combinations collapse those arrangements into one group. nCr = nPr ÷ r!.
Input constraints
Inputs should be whole numbers. Very large values can exceed practical display limits, so use realistic ranges for planning tasks.
Interesting facts
Order is ignored
Team {A,B,C} is the same combination as {C,B,A}; only membership matters.
Pascal triangle connection
Each binomial coefficient in Pascal’s triangle is a combination value nCr.
Combinations grow quickly
Even moderate values can be large, such as 52C5 = 2,598,960 poker hands.
Frequently asked questions
Combination ignores order; permutation counts order. Use combinations for group selection, permutations for ranking or arrangement.
Use nCr when choosing r items from n without repetition and without caring about order.
Use repetition mode when the same option can appear multiple times in one selection.
No. Standard combinations require n ≥ r.
No. Combination counting uses non-negative integers.
References
- Combinations — MathWorld Definition of combinations C(n, k) without regard to order.
- Counting and probability Introductory combinatorics for permutations and combinations.