Formula Used
Permutation: nPr = n! / (n - r)!. Use it when order matters.
Combination: nCr = n! / (r! × (n - r)!). Use it when order does not matter.
Permutation with repetition: n^r. Use it when choices can repeat and order matters.
Combination with repetition: (n + r - 1)! / (r! × (n - 1)!). Use it when choices can repeat and order does not matter.
Circular permutation: (n - 1)!. Use it when rotations count as the same arrangement.
Multiset permutation: n! / (a! × b! × c! × ...). Use it when duplicate objects exist.
Derangement: !n = (n - 1) × (!(n - 1) + !(n - 2)). Use it when no item stays in place.
How to Use This Calculator
- Choose the calculation type.
- Enter n as the total number of available items.
- Enter r as the number selected or arranged.
- For multiset mode, enter duplicate counts like 2,2,1.
- Press the calculate button.
- Review the exact answer, formula, digit count, and scientific form.
- Use the CSV or PDF button to save the result.
Example Data Table
| Case |
Input |
Formula |
Result |
| Permutation |
n = 5, r = 3 |
5P3 |
60 |
| Combination |
n = 5, r = 3 |
5C3 |
10 |
| Permutation with repetition |
n = 4, r = 3 |
4^3 |
64 |
| Combination with repetition |
n = 4, r = 3 |
6C3 |
20 |
| Circular permutation |
n = 6 |
5! |
120 |
| Multiset permutation |
2,2,1 |
5! / (2! × 2! × 1!) |
30 |
| Derangement |
n = 5 |
!5 |
44 |
About This Calculator
Counting problems often look simple. They can still confuse teams. The main question is order. If order matters, use a permutation. If order does not matter, use a combination. This calculator handles both ideas. It also supports repeated picks, circular seating, grouped duplicate items, and derangements.
Why It Helps
Manual factorial work grows very fast. A small input can create a huge answer. This tool keeps the steps visible. It reports the exact value. It also shows a compact scientific form. That makes large answers easier to read. Students can compare methods. Writers can check examples. Analysts can document assumptions before sharing results.
Common Counting Choices
Use nPr when you arrange selected items. Race results are a good example. First, second, and third places are ordered. Use nCr when you choose items without rank. A committee selection is a common example. Use repetition when the same item can be picked again. Password patterns and product codes often need that setting. Use multiset mode when some items are identical. Letter arrangements are easier with this option.
Practical Notes
Always define n as the total available items. Define r as the number selected or arranged. Keep both values as whole numbers. Negative values do not fit these formulas. For circular arrangements, rotations are treated as the same layout. For derangements, no item may stay in its original position.
Use Results Carefully
Counting answers depend on the model. A tiny rule change can change the result. Replacement, order, and identical objects must be decided first. The exported file helps record those choices. Use the table for quick checking. Then compare your own problem with the closest example. This habit prevents common counting errors. It also makes reports clearer for readers.
For learning, change one input at a time. Notice how the answer moves. Large jumps are normal. Factorials multiply many terms. Repetition can grow even faster. Clear labels make the final number easier to trust and reuse later.
Save exports with each finished calculation.
Final Tip
Start with the story behind the problem. Ask whether positions matter. Ask whether items repeat. Ask whether duplicates exist. Then choose the matching method. The calculator will finish the arithmetic and show the formula.
FAQs
What is the difference between combinations and permutations?
Permutations count arrangements where order matters. Combinations count selections where order does not matter. Choosing three winners by rank uses permutations. Choosing three team members uses combinations.
What does n mean?
n is the total number of available items. It may represent people, digits, products, letters, seats, or any set of objects used in the counting problem.
What does r mean?
r is the number of items selected or arranged from n. In 10P3 or 10C3, r is 3. It must be a whole number.
When should I use repetition mode?
Use repetition mode when the same item can be used more than once. Password digits, PIN codes, and repeated product choices often need this setting.
When should I use circular permutation?
Use circular permutation when arrangements around a circle are counted. Rotating the same seating order does not create a new arrangement in this method.
What is a multiset permutation?
A multiset permutation counts arrangements when duplicate items exist. For example, the letters in a word may repeat. Duplicate groups reduce the total arrangement count.
What is a derangement?
A derangement is an arrangement where no item remains in its original position. It is useful for secret swaps, envelope problems, and misplacement models.
Why are some answers very large?
Factorials grow quickly. Even moderate n values can create huge numbers. The calculator shows exact values and scientific form to make large results easier to inspect.