Permutation and combination calculator with repetition
Enter n, r and whether choices may repeat. HitEquals reports the ordered count, unordered count, exact integers and digit lengths without converting large results to floating point.
Inputs used by this calculator
Permutation and Combination Calculator uses the fields below. The starting values reproduce the page example; replace them with values from the same units and time period before using the result.
| Input | Starting example | Accepted value | How it is used |
|---|---|---|---|
| Available items (n) | 10 | 1 to 500 | Supplies the available items (n) quantity used by the published equation or result breakdown. |
| Items selected (r) | 3 | 0 to 500 | Supplies the items selected (r) quantity used by the published equation or result breakdown. |
| Repetition | Not allowed | Not allowed · Allowed | Supplies the repetition quantity used by the published equation or result breakdown. |
Permutation versus combination
A permutation counts arrangements where order matters. A combination counts selections where order does not matter.
Without repetition
Once an item is selected it cannot be selected again. This requires r to be no greater than n.
nPr=n!/(n−r)!nCr=n!/[r!(n−r)!]With repetition
For ordered strings each position can use any of n items. Unordered multisets use the stars-and-bars combination formula.
ordered=n^runordered=C(n+r−1,r)Worked 10 choose 3 example
There are 10×9×8=720 ordered selections. Dividing by 3!=6 removes the internal order and leaves 120 combinations.
Why exact integers matter
Factorials grow quickly. The engine uses BigInt so it does not round large whole-number counts to scientific approximations.
Limits and interpretation
The calculation assumes distinct available items and a fixed repetition rule. Identical objects, adjacency restrictions or conditional choices need specialized counting methods.
Four counting models
| Order | Repetition | Formula |
|---|---|---|
| Matters | No | n!/(n−r)! |
| Does not matter | No | n!/[r!(n−r)!] |
| Matters | Yes | n^r |
| Does not matter | Yes | C(n+r−1,r) |
Frequently asked questions
When should I use a permutation?
When changing the order creates a different outcome.
When should I use a combination?
When only membership matters, not order.
Can r be zero?
Yes. There is one empty selection.
Can r exceed n?
Only when repetition is allowed.
Are large answers exact?
Yes. Counts use integer BigInt arithmetic.
Does the tool list every arrangement?
No. It calculates the number of arrangements.
What to keep in mind
Items are treated as distinct before selection. The calculator returns exact integers up to n and r of 500; it does not enumerate or store the arrangements themselves.
Results are rounded for display in US format, with money in dollars and cents; CSV exports keep full calculation precision. This is a calculation summary, not an official certificate.