Team selection
How many ways can you choose 3 speakers from 10 candidates? This is a classic combinations problem because only the final group matters, not the order you listed them in.
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Combinations Calculator
Calculate unordered selections with and without repetition using exact combinatorics formulas. Useful for probability setup, sampling, menu choices, team selection, bundle planning, and classroom math problems.
Combinations Calculator
Use this free combinations calculator to find how many unordered selections are possible from a larger set. It supports standard combinations without repetition and combinations with repetition, making it useful for probability, statistics, sampling, inventory choices, classroom combinatorics, and planning problems. If you want to connect the result to event likelihood next, continue with the Probability Calculator, or browse more tools in Math & Statistics Calculators.
Count unordered selections instantly
Enter n items and r selections to calculate how many unique groups are possible when order does not matter.
Useful for
Sampling and choice counts
Great for teams, menus, card hands, bundles, and probability setup.
Inputs
Use whole numbers only. In standard mode, n is the total number of distinct items and r is how many you choose.
Example: choosing 3 people from 10 uses n = 10 and r = 3.
Combination count
120
Standard combinations count unordered selections without reuse. The same group is counted once no matter how you arrange it.
Formula used
nCr = n! / (r! × (n - r)!)
Exact result
120
If order mattered
720
Calculation steps
- 1Start with n = 10 and r = 3
- 2Use the formula nCr = n! / (r! × (n - r)!)
- 3Here that becomes 10! / (3! × (10 - 3)!)
- 4The exact combinations result is 120
- 5If order mattered instead, the permutations count would be 720
Interpretation
This result tells you how many unique groups are possible once duplicate ordering is removed. That is why combinations are often used before moving into probability, lottery odds, sample design, product bundle planning, or card-hand analysis.
What this combinations calculator does
This tool counts how many distinct selections are possible when the order of the chosen items does not matter. That makes it useful for committee selection, menu choices, bundle construction, sample picking, and many probability setup problems.
How combinations work
A combinations formula removes duplicate arrangements that represent the same group. For standard combinations, nCr divides factorial-based arrangements by the number of internal orderings. For repetition cases, the formula shifts to C(n + r - 1, r).
Combinations formulas explained
Without repetition
nCr = n! / (r! × (n - r)!)
With repetition
C(n + r - 1, r)
n
Total distinct items available.
r
Number of items selected.
nCr
Number of unordered selections without repetition.
nPr
Permutation count when order matters.
Combinations calculator examples
Menu or product bundles
If customers can choose a fixed number of options from a larger list, combinations tell you how many distinct bundles or menus are possible.
Probability setup
Lottery problems, card hands, and sample selection often begin with combinations. After counting possible groups, use the Probability Calculator to turn those counts into actual event likelihoods.
If you need to convert this count into event odds next, continue with the Probability Calculator. If you are building a broader data workflow, the Standard Deviation Calculator and Mean Calculator help cover spread and averages after your combinatorics setup is complete.
Why combinations matter
Combinations are the right tool whenever you care about which items were selected but not the order they appear in. That distinction prevents overcounting and makes later probability work accurate.
When repetition changes the answer
Allowing the same choice more than once increases the number of possible groups. That is why combinations with repetition use a different formula than standard nCr.
Math & statistics workflow
Combinations often sit upstream of probability, sampling, and statistical modeling. They help define the size of a choice space before you analyze outcomes, averages, or variability.
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Frequently asked questions
It calculates how many unordered selections are possible when choosing r items from n total items. Because order does not matter in combinations, choosing A, B, C is the same as choosing C, B, A.
Explore This Tool in Context
Combinations Calculator is part of the Math & Statistics Calculators collection. If you want a broader view of similar workflows, open the Math & Statistics Calculators category page or browse all QuickTools categories.
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