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Formula verified · Deterministic calculation

Combination Calculator

Enter n and r to calculate the number of combinations when selection order does not matter.

Combinations10
Total objects (n)5
Selected objects (r)3
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Result methodology

Exact unordered selections without replacement using C(n,r)=n!/(r!(n−r)!).

Solution steps
  1. n = 5; r = 3
  2. C(n,r) = n!/(r!(n-r)!) = 10

How this calculator works

Combinations use C(n,r)=n!/(r!(n−r)!).

Formula

C(n,r)=n!/(r!(n−r)!)

Counts unordered selections of r objects from n distinct objects without replacement.

Worked example

Five choose three is a verified test case used by the calculation engine.

Assumptions and limitations

  • n and r are non-negative integers with r≤n.
  • Objects are distinct and selections are made without replacement.
  • Results are limited to the supported exact integer range.

Methodology & sources

This calculator uses deterministic, versioned calculation logic. The formula and verified examples above are part of the calculation definition used by CalcuMint.

  • CalcuMint deterministic statistics engine

Frequently asked questions

How are combinations different from permutations?

Combinations ignore selection order; permutations count different orders separately.

What is C(n,0)?

There is exactly one way to select zero objects, so C(n,0)=1.

Related calculators

Permutation CalculatorCalculate ordered selections of r objects from n distinct objects.Binomial Probability CalculatorCalculate the probability of exactly k successes in n independent trials.Probability CalculatorCombine two event probabilities under an explicit independence assumption.