Combinatorics Variants Calculator
Count arrangements beyond basic nPr/nCr: permutations with replacement (nʳ), combinations with replacement (C(n+r−1, r)), and circular permutations ((n−1)!).
Inputs explained
- Variant — Which counting rule to apply.
- n (total items) — Number of distinct items to choose from. Whole number of 1 or more.
- r (chosen items) — How many items are chosen (not used for circular permutations).
How to use it
- Fill in the fields above — each input has a hint describing exactly what to enter
- Press Calculate to get your result instantly, right on this page
- Copy the result, save the tool to favorites, or share and embed it anywhere
Understanding the result
Example: With n = 5 and r = 3: permutations with replacement = 125, combinations with replacement = 35, and 5 people around a circular table can sit in 24 ways.
Frequently asked questions
How is 'permutations with replacement' different from a regular permutation (nPr)? Regular nPr assumes each item can only be picked once, so the pool shrinks with each choice; permutations with replacement (nʳ) allow the same item to be picked again for every position, which is why the count is a simple power of n.
Why does the circular permutation formula use (n−1)! instead of n!? In a circular arrangement, rotating everyone one seat over is considered the same arrangement, so one position is effectively fixed as a reference point, leaving only (n−1) items to arrange freely.
Is the Combinatorics Variants Calculator free to use? Yes — it is completely free, with no sign-up, no download and no usage limits. You can use it as often as you need.
Is my data private? Yes. The calculation runs entirely in your browser — the values you enter are never sent to our servers, stored or shared with anyone.
What information do I need? Just the following: Variant, n (total items), r (chosen items). Each field includes a hint explaining exactly what to enter.
Can I use it on my phone? Yes — the tool is fully responsive and works on any device, and the same calculator is available in the EasyWorkCalcs mobile app.
Privacy
Everything you enter into the Combinatorics Variants Calculator is processed entirely in your browser. Nothing is uploaded, stored or shared — your data stays on your device.