Example Data Table
| Case |
Inputs |
Formula |
Answer |
| Ranked prizes |
n = 8, r = 3 |
P(8,3) |
336 |
| Committee choices |
n = 8, r = 3 |
C(8,3) |
56 |
| Four digit code |
n = 10, r = 4 |
10^4 |
10000 |
| Round table seating |
n = 6 |
(6-1)! |
120 |
| Repeated letters |
counts = 2,2,1 |
5! / (2!2!1!) |
30 |
| No fixed item |
n = 6 |
!6 |
265 |
Formula Used
Permutation without repetition: P(n,r) = n! / (n-r)!. Use it when order matters and items cannot repeat.
Combination without repetition: C(n,r) = n! / (r!(n-r)!). Use it when order does not matter.
Permutation with repetition: n^r. Use it for codes, passwords, and ordered picks where repeats are allowed.
Combination with repetition: C(n+r-1,r). Use it for unordered repeated selections.
Circular arrangement: (n-1)! for all distinct items around a circle. For r chosen from n, use n! / ((n-r)!r).
Multiset permutation: n! / (a!b!c!...). Use it when identical items appear.
Derangement: !n = (n-1)(!(n-1)+!(n-2)). Use it when no item keeps its original position.
How to Use This Calculator
- Select the counting model that matches your problem.
- Enter n as the total number of available items.
- Enter r as the number of selected items, when needed.
- For multiset problems, enter repeated group counts like 2,2,1.
- Add an optional note for your exported report.
- Press Calculate to show the result above the form.
- Use CSV for spreadsheets or PDF for a clean document.
Counting Choices with Confidence
Permutation and combination work appears in exams, games, staffing, security codes, lotteries, schedules, and quality checks. The ideas look simple. Yet small changes can change the answer. Order may matter. Repeated items may be allowed. Objects may sit around a circle. Some items may be identical. This calculator keeps those cases separate. It also shows the formula, inputs, digit count, and exportable answer.
Why Order Matters
A permutation counts arrangements. The sequence ABC is different from BAC. That is why passwords, race finishes, seating lists, and ranked winners often use permutations. A combination counts groups. The group ABC is the same as BAC. That is why committees, card hands, sample groups, and bundle choices often use combinations. Choosing the correct model is the most important step.
Useful Advanced Cases
Repeated selections are common. A PIN can reuse digits. A menu order can reuse options. The tool handles repeated permutations with n to the power r. It also handles repeated combinations with the stars and bars rule. Circular arrangements remove rotations that look the same. Multiset arrangements divide by repeated item factorials. Derangements count cases where no item stays in its original position. These cases support deeper counting tasks.
Better Planning and Review
The calculator is built for clear review. Enter n for available items. Enter r when you choose a smaller set. Use the identical counts field for repeated labels, such as 2,2,3. Submit the form to see the result above the inputs. The exact value stays visible for copying. The summary helps when the number is long. CSV export is useful for spreadsheets. PDF export is useful for homework, reports, and audit notes.
Practical Accuracy Tips
Always define the experiment first. Ask if order matters. Ask if repeats are allowed. Ask if rotations are identical. Ask if some objects are identical. Then pick the matching option. Keep n and r as whole numbers. Make sure r is not greater than n for non-repeating choices. Use examples to verify the model. A small test can prevent a large counting mistake. Document each assumption beside the result. This habit makes later checking easier. It also helps teachers, clients, and teammates understand the answer with confidence.
FAQs
1. What is the difference between permutation and combination?
A permutation counts ordered arrangements. A combination counts unordered groups. Use permutation when sequence changes the outcome. Use combination when the same selected items create one group.
2. When should I use repetition?
Use repetition when an item can be chosen more than once. Passwords, PINs, menu choices, and replacement sampling often allow repeated selections.
3. What does n mean?
n is the total number of available items. For example, if you choose from eight students, then n equals 8.
4. What does r mean?
r is the number of chosen positions or selected items. For example, choosing three winners from eight people means r equals 3.
5. What is a circular arrangement?
A circular arrangement counts orders around a circle. Rotations are treated as the same arrangement, so the basic formula becomes (n-1)!.
6. How do multiset permutations work?
Multiset permutations handle identical items. First count all items. Then divide by the factorial of each repeated group count.
7. What is a derangement?
A derangement counts arrangements where no item stays in its original position. It is useful for shuffled assignments and matching problems.
8. Can I export the answer?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for printable reports, homework records, and sharing.