Combinations Calculator
Calculate how many ways you can choose r items from n without regard to order.
Last updated · View algorithm, constants and default inputs (JSON)
About this calculator
Combinations count selections where order does not matter — a hand of cards rather than a sequence of them. The formula is the binomial coefficient, and it appears throughout probability and statistics.
Worked example
calculated with the default values below
Total items (n) 10 · Items chosen (r) 3 Combinations 120 · Permutations 720 · Combinations with repetition 220
General background references
Disclaimer
These calculators give general mathematical results. They suit study, planning and estimation — verify anything that feeds a graded submission, a published figure or a financial commitment.
Implementation source
This record describes the implementation used by this page. It is not independent validation. External references, when available, support only the stated formula or definition; estimates and custom rules retain their own assumptions.
Formula and variables
C(n,r) = n! ÷ (r!(n−r)!); P(n,r) = n! ÷ (n−r)!; C_rep = (n+r−1)! ÷ (r!(n−1)!)
- n
- total items
- r
- items chosen
- C(n,r)
- combinations
- P(n,r)
- permutations
- C_rep
- combinations with repetition
Inputs must be whole numbers from 0 to 100. Without repetition, results are 0 when r > n; combinations with repetition are calculated separately. Counts are returned as exact integer text.
Frequently asked questions
Why are combinations with repetition larger than plain combinations?
Plain combinations forbid reusing an item, so choosing 3 from 10 gives 120 ways. With repetition the formula becomes (n + r − 1) choose r, which for 10 and 3 is (12 choose 3) or 220 — nearly double. Repetition lets the same item appear several times, widening the pool without raising n. The permutations output stays separate because order then matters. Without repetition, choosing more items than the set contains gives zero combinations and zero permutations. The with-repetition result is calculated separately, so it can still be positive when r is greater than n.