Permutations & Combinations Calculator
See how many distinct outcomes you can make when order does or does not matter. This tool shows counts for permutations (arrangements) and combinations (selections), with and without repetition, so you can size scenarios like seating charts, password or code spaces, lottery-style draws, product bundles, and experiment designs at a glance.
The Permutations & Combinations Calculator lets you enter set sizes and pick counts to compute totals, compare cases (order vs. no order), and translate results into probabilities when needed. The goal is quick, error-free reasoning about “how many ways?” so you can plan trials, estimate odds, and communicate complexity clearly-without getting lost in factorials.
Enter n and r to get permutations (nPr) and combinations (nCr) - with or without repetition - plus steps, formulas, and examples.
Results interpretation
- Permutations count arrangements where order matters.
- Combinations count selections where order doesn’t matter.
- Check “repetition allowed” if items can repeat (e.g., PIN codes). Leave unchecked for draws without replacement.
- Large n and r produce very large counts-shown in compact scientific notation when needed. Extremely large inputs may exceed standard numeric limits.
How this calculator works
Formula & assumptions
No repetition: nPr = n! / (n − r)! and nCr = n! / (r!(n − r)!). We compute nPr as a direct product and nCr using a multiplicative formula that stays numerically stable for typical inputs.
With repetition: Permutations = n^r and Combinations = C(n + r − 1, r).
Assumptions: all items are distinct; each selection step is independent; there are no additional pattern-based constraints (like “no adjacent vowels”).
Use cases & examples
Example 1 (no repetition): How many 3-letter codes from 10 letters with no repeat? Permutations: nPr = 10×9×8 = 720. Combinations (ignoring order): nCr = C(10,3) = 120.
Example 2 (with repetition): 4-digit PIN codes from 10 digits: n^r = 10^4 = 10,000. Combinations with repetition for picking 4 scoops from 10 flavors: C(10+4−1,4) = C(13,4) = 715.
Example 3 (cards): 5-card poker hands: C(52,5) = 2,598,960 (order doesn’t matter).
Permutations & Combinations - FAQ
When do I choose permutations vs. combinations?
Use permutations when order matters; combinations when order doesn’t.
What does “repetition allowed” mean?
Items can be reused (like digits in a PIN). Without repetition, each item can be used at most once.
Why is nCr smaller than nPr?
Combinations divide out the r! re-orderings of the same selection.
Can r exceed n?
Only when repetition is allowed. Without repetition, r must be less than or equal to n.
Why do the numbers get so large?
Counting grows rapidly with n and r. The calculator uses standard floating-point arithmetic and shows compact scientific notation for very large values.
Do these formulas assume all items distinct?
Yes. If you have repeated or identical items with extra constraints, the counts may differ and require a custom derivation.
Permutations vs. Combinations: A Practical Guide for Counting
Our permutations and combinations calculator helps you answer a deceptively simple question: “How many ways can this happen?” Whether you’re exploring PIN codes, designing an experiment, building a sampling plan, or sizing a security keyspace, the key distinction is whether order matters and whether repetition is allowed. When order matters, you’re counting arrangements-that’s permutations. When order doesn’t, you’re counting selections-that’s combinations.
In the classic “no repetition” setting, permutations count all r-long strings drawn from n distinct items without reuse: nPr = n!/(n − r)!. Combinations collapse order by dividing out the r! ways to reorder the same selection: nCr = n!/(r!(n − r)!). If you allow repetition, permutations become n^r (each position picks from all n options), and combinations become the “stars and bars” formula: C(n + r − 1, r), which counts multisets of size r from n types.
Getting the switches right is more than a textbook exercise. In quality engineering, combinations quantify how many test cases cover r-way interactions across n factors. In cybersecurity, permutations with repetition approximate password keyspaces under policy constraints. In product, combinations explain why a seemingly small set of options explodes into thousands of configurations. And in research, combinations determine how many sample draws are possible or how many committee selections exist from a pool.
Large inputs generate astronomically large integers in theory. This tool uses standard floating-point math and displays a readable scientific shorthand (e.g., 1.23e+45) when needed. Two practical tips: (1) sanity-check that inputs reflect the real rules-can items repeat? does order matter? (2) if a scenario has additional constraints (e.g., “no adjacent vowels”), the count must incorporate those rules and may require a custom derivation.
With the right model, the counting falls out immediately. Use permutations for ordered arrangements, combinations for unordered selections, flip the repetition toggle as your scenario allows, and you’ll have a clean, defensible answer in seconds.