Combinations without repetition calculator
Count unordered selections of r distinct objects chosen from n distinct objects, and generate the actual combinations when the set has at most 10 objects.
Inputs
Results
Combination generator
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How to use this combinations without repetition calculator
What this calculator does
This calculator finds how many unordered groups of a fixed size can be selected from a collection of distinct objects when an object may be used at most once. In combinatorics this count is the binomial coefficient, written C(n,r) or “n choose r.” It answers a counting question, not a probability question by itself, and it does not treat different orders of the same selected objects as different outcomes. For example, selecting objects 1, 2, 3, and 4 is the same combination as selecting 4, 3, 2, and 1.
When to use it
Use the calculator when choosing a committee from a larger group, counting possible card hands, selecting a fixed number of product options without duplicates, or finding the denominator of a probability model where every unordered subset is equally likely. If order matters, the problem is a permutation instead. OpenStax gives practical examples of this distinction in its combinations lesson.
How to calculate
The calculator opens with a ready-to-use demonstration: n = 10 and r = 4. Its results and a validated Excel workbook are available immediately. To use your own values:
- Replace Total number of objects n with the number of distinct objects available.
- Replace Sample size r with the number of objects selected in each group. Keep r between 0 and n.
- Read Combinations without repetition for the exact number of unordered selections. The smaller cards show Objects not selected and Generated rows.
- If n is 10 or less, inspect the Combination generator table. Each row lists one unique selection using labels 1 through n.
- Select Download Excel to export the current inputs, result, formula, and generated list when available. Reset clears both numeric inputs, clears computed content, and disables export until a complete valid pair is entered again.
Input guide
Total number of objects n is a required nonnegative integer representing the size of the distinct source set. Plain digits such as 10 and correctly grouped integers such as 1,000 are accepted. Decimal values, signs, scientific notation, and malformed grouping are rejected. Increasing n while keeping r fixed normally increases the number of possible combinations. A common mistake is counting duplicated objects as distinct when they are not actually distinguishable.
Sample size r is a required nonnegative integer from 0 through n. The startup example uses 4. Setting r to 0 gives one combination – the empty selection – and setting r equal to n also gives one combination – the entire set. A common mistake is entering a sample size greater than the number of available objects; that is invalid when repetition is forbidden.
Output guide
Combinations without repetition is the exact count of unordered r-element subsets. It is displayed as an integer and can be very large. Objects not selected equals n – r and helps interpret the complementary choice. Generated rows reports how many rows are currently materialized in the generator: it equals the exact count when n ≤ 10, and is zero for larger sets because enumeration is intentionally suppressed. The formula line shows the current binomial-coefficient identity. The header pills repeat n and r and indicate whether the generator is available; they are status summaries, not additional calculations.
The Combination generator table has two columns: # is the sequential row number, while Combination lists selected labels. The table is an exact enumeration only for n ≤ 10. A zero generated-row count for n > 10 does not mean there are zero combinations; it means only that the listing is omitted while the exact count remains available.
Worked example
With the startup values n = 10 and r = 4, the calculation is C(10,4) = 10! / (4! × 6!) = 210. There are exactly 210 unordered ways to select four distinct objects from ten. Six objects remain unselected in each choice, so Objects not selected is 6. Because n is within the generator limit, all 210 selections are listed and Generated rows is 210. The first generated row is 1, 2, 3, 4, and the last is 7, 8, 9, 10.
Learn more
The underlying formula and meaning of “unordered selection” are also documented in Wolfram MathWorld's combination reference.
How the combinations formula works
For nonnegative integers with 0 ≤ r ≤ n, the count is C(n,r) = n! / (r!(n-r)!). One way to understand the division is to start with ordered selections and divide by r!, because each unordered group can be internally rearranged in r! ways without changing which objects were selected. The symmetry identity C(n,r) = C(n,n-r) follows naturally: choosing which r objects to include is equivalent to choosing which n-r objects to leave out.
For reliable large-integer arithmetic, this calculator evaluates the binomial coefficient with a multiplicative integer formula rather than first constructing enormous factorials. The result stays exact instead of using floating-point approximations. MathWorld's binomial coefficient reference describes the same quantity and its standard notation.
Common interpretation mistakes
- Using combinations when order matters. A committee of three is usually a combination; first, second, and third place is a permutation.
- Allowing repeats accidentally. This calculator assumes each source object can appear at most once in a selected group.
- Confusing count with probability. The number of combinations can be an ingredient in a probability calculation, but probability also requires a model for which outcomes are possible and how likely they are.
- Interpreting an omitted list as zero. For n above 10, only the enumeration is omitted; the exact binomial coefficient is still calculated.