Find nCr — combinations of n items chosen r at a time — with or without repetition, computed exactly using BigInt arithmetic.
Enter n and r and click Calculate to see the result.
This free combination calculator finds nCr — also called "n choose r" or the binomial coefficient — the number of ways to select r items from a set of n where the order of selection doesn't matter, with exact results computed using BigInt arithmetic so even large factorials stay precise. It also covers combinations with repetition, where the same item can be chosen more than once. Whether you're working out lottery odds, a poker-hand probability, or a committee-selection problem, this combination calculator with steps shows the exact factorial substitution behind every answer.
A combination counts unordered selections: choosing r items out of n where swapping the order two items were picked in does not create a new outcome. This calculator computes the "without repetition" case (nCr = n!/(r!(n−r)!), the classic binomial coefficient) and the "with repetition" case (C(n+r−1, r), for when the same item can be picked more than once).
Students and teachers working through combinatorics and probability coursework, statisticians computing binomial probabilities, lottery and card-game odds analysts, quality-control engineers sampling from a batch, and anyone who needs an exact count of unordered selections can use this tool.
Many real problems only care about which items end up selected, not the order they were chosen in. A 5-card poker hand, a lottery ticket's set of numbers, or a committee formed from a larger group are all combination problems, not permutation problems, because rearranging the same selected items doesn't create a new outcome. The binomial coefficient nCr also appears throughout probability theory, most famously in the binomial distribution.
Lottery analysts compute the odds of matching a winning combination. Card players calculate the number of possible poker hands. Statisticians use the binomial coefficient directly in binomial probability formulas. Quality control teams calculate how many ways to sample a batch of items for inspection. Committee and team-selection problems in schools and organizations rely on the same math.
How this calculator counts unordered selections
Combinations treat different orderings of the same selected items as identical — choosing {A,B,C} is the same combination regardless of pick order.
Every combination corresponds to r! different permutations, so nCr = nPr ÷ r! — dividing out the orderings a combination ignores.
nCr is the coefficient of each term when expanding (x+y)^n, which is why it's central to Pascal's Triangle and the binomial theorem.
From choosing a mode to reading the exact count
Pick "Without Repetition" (the standard nCr case) or "With Repetition" depending on your problem.
Type the total number of items (n) and how many you're choosing (r).
The calculator applies the matching combination formula instantly using exact BigInt arithmetic.
See the exact combination count, plus the factorial substitution used to reach it.
For small n and r, manually list a few selections to spot-check the calculator's count.
Choosing 3 topping flavors out of 10 available
A dessert shop offers 10 topping flavors. In how many distinct ways can a customer choose 3 of them, if the order they're picked in doesn't matter?
Explanation: There are 120 distinct topping selections, because {vanilla, chocolate, mint} counts the same regardless of pick order. Compare this to the permutation count of 720 for the same n and r — the combination count is exactly 720÷3!=720÷6=120, since each unordered selection of 3 flavors corresponds to 3!=6 different orderings.
What the result represents
| Mode | What the Result Means |
|---|---|
| Without Repetition | The exact count of unordered selections of r items chosen from n, with no item reused |
| With Repetition | The exact count of unordered selections of r items where any of the n items can be chosen more than once |
Typical ranges: combination counts grow quickly but always more slowly than the corresponding permutation count for the same n and r, since nCr = nPr ÷ r!.
Manual verification: for small n and r, list out the distinct unordered selections by hand, or check that nCr equals nC(n−r) (the symmetry property) as a quick sanity check.
Where counting unordered selections matters
Check combination word problems step by step, with the exact factorial shown.
Compute the total number of possible ticket combinations.
Count possible poker hands or card-draw combinations.
Use nCr directly as the binomial coefficient in probability formulas.
Count the number of ways to form a group from a larger pool.
Calculate how many ways to select a sample batch for inspection.
Count possible topping, flavor, or add-on selections.
Analyze subset-selection and combinatorial search problems.
Count possible gene or allele combinations in a sample.
Practice the combination problems common in SAT/GRE/GMAT quantitative sections.
Explore the coefficients that appear when expanding (x+y)^n.
Count possible prize-winner selection combinations.
What this combination calculator does well, and where it doesn't apply
The single question that tells them apart: does order matter?
| Concept | Order Matters? | Formula | Example (n=10, r=3) |
|---|---|---|---|
| Combination (nCr) | No | n!/(r!(n−r)!) | 120 |
| Permutation (nPr) | Yes | n!/(n−r)! | 720 |
Summary: This combination calculator gives you an instant, free, exact way to count unordered selections — with or without repetition — using BigInt-precision factorial math. Pair it with the Permutation Calculator when order matters, and the Factorial Calculator to check the underlying factorial values.
Common questions about combinations
Trusted educational references to go deeper on combinatorics
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