This professional permutation and combination calculator gives exact results using big-integer arithmetic. It supports permutations and combinations, with or without repetition, and provides a clear step-by-step explanation, formulas, and trustworthy references. Ideal for students, educators, data scientists, and exam preparation.
Data Source and Methodology
Primary reference: NIST Digital Library of Mathematical Functions (DLMF), Chapter 26: Combinatorial Functions, Release 1.1.11 (2023-06-15). https://dlmf.nist.gov/26
Secondary reference: Graham, Knuth, and Patashnik, Concrete Mathematics, 2nd Edition, Addison-Wesley (1994).
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
The Formula Explained
Permutations without repetition: \( P(n,r) = {}_nP_r = \dfrac{n!}{(n-r)!} = n \times (n-1) \times \cdots \times (n-r+1) \)
Permutations with repetition: \( P_{\text{rep}}(n,r) = n^r \)
Combinations without repetition: \( C(n,r) = {}_nC_r = \dfrac{n!}{r!(n-r)!} \)
Combinations with repetition: \( C_{\text{rep}}(n,r) = \binom{n+r-1}{r} = \dfrac{(n+r-1)!}{r!(n-1)!} \)
Glossary of Variables
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- n: Number of distinct available items (integer ≥ 0).
- r: Selection size, how many items are chosen (integer ≥ 0).
- Permutation (nPr): Count of ordered selections from n items taken r at a time without repetition.
- Permutation with repetition: Ordered selections where items may repeat, equal to n^r.
- Combination (nCr): Count of unordered selections from n items taken r at a time without repetition.
- Combination with repetition: Unordered selections allowing repeats, equal to C(n+r−1, r).
Worked Example
How it works: a step-by-step example
Question: How many 3-character codes can be made from 10 distinct symbols if no symbol can be reused?
Because order matters and repetition is not allowed, use permutations without repetition:
\( P(10,3) = \dfrac{10!}{(10-3)!} = 10 \times 9 \times 8 = 720 \)
Therefore, there are 720 possible codes.
Frequently Asked Questions (FAQ)
1) How do I choose between permutations and combinations?
If order matters (e.g., PIN codes, rankings), use permutations. If order does not matter (e.g., choosing a hand of cards), use combinations.
2) What changes when repetition is allowed?
With repetition, items can be chosen multiple times. For permutations it becomes \( n^r \); for combinations it becomes \( \binom{n+r-1}{r} \).
3) Can I enter very large n and r?
The calculator uses exact big-integer math for correctness. For performance and usability, inputs are limited to reasonable sizes (default: up to 1000). This cap avoids slow computations and can be adjusted in code.
4) Do you show exact and approximate results?
Yes. The main figure is exact. We also show an approximate scientific notation and the number of digits for quick interpretation.
5) Why is r ≤ n required when repetition is off?
Without repetition, you cannot pick more items than available. Enable repetition to allow r > n scenarios.
6) Are there edge cases?
Common identities: \( C(n,0)=1 \), \( C(n,n)=1 \), \( P(n,0)=1 \), \( n^0=1 \). The tool handles these automatically.
7) Where can I learn more?
See the NIST DLMF Chapter 26 and Concrete Mathematics (Graham–Knuth–Patashnik) for authoritative coverage of combinatorial functions.