Permutations and combinations are counting methods used in probability, mathematics, scheduling, and decision-making. The key distinction is whether the order of selected items matters. A permutation and combination calculator can help evaluate the formulas efficiently.
Permutation vs combination
A permutation is an arrangement in which order matters. A combination is a selection in which order does not matter. Choosing a president and vice president from a group is an ordered assignment; choosing two members for a committee is not.
How to use the calculator
- Open the Permutation and Combination Calculator.
- Enter n, the total number of distinct available items.
- Enter r, the number of items to select or arrange.
- Select permutation or combination if the tool provides this option.
- Calculate and confirm whether order matters for your actual problem.
Permutation formula
When selecting r distinct items from n distinct items without replacement and order matters, the formula is nPr = n! / (n − r)!. For example, assigning first, second, and third place among 5 finalists gives 5P3 = 5 × 4 × 3 = 60 possible ordered outcomes.
Combination formula
When selecting r distinct items from n distinct items without replacement and order does not matter, the formula is nCr = n! / [r!(n − r)!]. Choosing 3 people from 5 for an unordered group gives 5C3 = 10 possible groups.
What does factorial mean?
Factorial, written n!, multiplies positive integers from n down to 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1.
Comparison examples
| Scenario | Method | Result |
|---|---|---|
| Choose 2 of 4 for a team | 4C2 | 6 |
| Assign 2 distinct roles from 4 | 4P2 | 12 |
| Choose 3 of 6 | 6C3 | 20 |
| Arrange 3 of 6 in order | 6P3 | 120 |
Why the two answers differ
For three distinct selected items, there are 3! = 6 possible orders. Permutations count those arrangements separately, while combinations count the group once. Therefore, nPr = nCr × r! for the standard without-replacement setting.
Important assumptions
These formulas assume distinct items and selection without replacement, with 0 ≤ r ≤ n. Problems that allow repeated selections or contain identical objects require different counting rules.
Common mistakes
- Using permutations when order is irrelevant.
- Using combinations for rankings or assigned roles.
- Forgetting that factorial grows quickly.
- Applying without-replacement formulas to problems where repetition is allowed.
Frequently asked questions
What is 5C2?
10.
What is 5P2?
20.
Does order matter in a lottery selection?
It depends on the lottery rules. If only the selected set matters, combinations are appropriate.
Why is 0! equal to 1?
Defining 0! = 1 makes factorial identities and counting formulas consistent.
Use the Permutation and Combination Calculator to check your counting problems.
Moiz Ahmad is a digital marketer and SEO expert based in Dera Ismail Khan, Pakistan. With over 6 years of experience in SEO, WordPress development, and digital strategy, he has helped 30+ businesses rank on Google’s first page. He is the founder of DigiTechPak and Moiz Blogger. Connect: LinkedIn | Twitter | Moiz Blogger