Last updated October 2, 2026 · Report an error
Permutations and combinations both count the number of ways to choose r items from a set of n, but they answer different questions — permutations count arrangements where order matters, while combinations count selections where it doesn't.
This calculator applies the correct factorial formula for whichever one you need and shows every step.
How it works
Enter n (the total number of items) and r (how many you're choosing), and select whether order matters. The calculator computes the relevant factorials and simplifies the result.
- Enter n (total items).
- Enter r (items chosen).
- Enter type.
- Click Calculate to see your results.
Examples
Choosing a committee vs. choosing officers
Choosing 3 people from 10 for an unordered committee is a combination (10C3 = 120), while choosing a president, vice president, and treasurer from the same 10 is a permutation (10P3 = 720), since the roles make order matter.
Who should use it
- Probability and statistics homework involving counting outcomes.
- Calculating the number of possible lottery combinations, passwords, or seating arrangements.
Industry applications
- Mathematics and statistics education
- Probability theory and combinatorics
Advantages
- Handles both permutations and combinations in one tool.
- Shows the full factorial substitution for transparency.
Common mistakes to avoid
- Using the combination formula when order actually matters (or vice versa), which gives a technically correct-looking but wrong answer to the actual question being asked.
- Forgetting that r cannot be greater than n — you can't choose more items than exist in the set.
Best practices
- Before calculating, ask "does the order of selection matter here?" — that single question determines whether you need a permutation or a combination.
Tips
- If you're unsure which to use, imagine swapping the order of two chosen items — if that creates a genuinely different outcome, you need a permutation; if not, use a combination.