Statistics
Combination & Permutation Calculator
Enter your details
Runs in your browser
How to use it
Using the combination & permutation calculator
- 01
Set n and r
n is how many items exist; r is how many you pick.
- 02
Read both answers
The main number is C(n,r); the permutation row gives P(n,r) for ordered scenarios.
- 03
Choose which applies
Lotteries, committees and hands are combinations; rankings, passwords and race finishes are permutations.
Good to know
Combinations or permutations?
Ask one question: would swapping two chosen items create a new outcome? A poker hand stays the same hand when reordered; combination. A podium finish changes meaning when gold and silver swap; permutation.
Why no factorial blowups
Directly computing 1000! overflows any floating-point format. Multiplying r ratios one at a time keeps every partial product a modest integer, which is why C(1000,3) computes instantly and exactly here.
- C(n,0) = C(n,n) = 1 always.
- P(n,r) = C(n,r) × r!
- r > n is rejected as impossible.
How it's calculated
The math behind this calculator
C(n,r) = n! / (r!(n−r)!) P(n,r) = n! / (n−r)!Combinations count unordered selections; permutations count ordered arrangements; same items, r! times as many results. Both are computed via running products rather than literal factorials, so intermediate values stay small and exact even for huge n where n! would overflow IEEE doubles (beyond 170!).
Validation enforces non-negative whole numbers with r ≤ n. Edge cases behave sensibly: choosing everything or nothing yields exactly one combination.
Assumptions & limitations
- n and r must be non-negative integers.
- Items are distinct; no repetition allowed within a selection.
- Order distinguishes permutations from combinations.
Worked example
A 5-card poker hand from a 52-card deck: 2,598,960 combinations; but 311,875,200 permutations once card order is considered.
FAQ
Frequently asked questions
- What if order does not matter but repeats are allowed?
- That is combinations with repetition: C(n+r−1, r); compute it here by substituting those adjusted arguments.
- How large can n be?
- Very large; the loop-product method avoids factorial overflow entirely. Only astronomically large exact results exceed floating-point integer precision beyond 2⁵³.
- Why does r ≤ n matter?
- You cannot choose more distinct items than exist; requesting r > n has no mathematical answer and returns an error.
- Is C(n,r) always an integer?
- Yes; it counts selections. The incremental division in the loop product preserves this exactly.
Keep exploring