Combination Calculator
Use this calculator when order does not matter — committee picks, lottery choices, product bundles, and menu pairings.
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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). Circular grouping contexts still use unordered selection rules unless seating order itself matters.
Worked example
- Standard combination with n = 5 and r = 3.
- nCr = 5!/[3!(5−3)!] = 5!/(3!·2!).
- 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 flavors 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.