Permutation & Combination Calculator

Calculate permutations (nPr), combinations (nCr), and factorials with step-by-step solutions.

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n (total items)

r (items chosen)

Mode

Combinations C(n,r)

Permutations P(n,r)

(n−r)!

Pascal's Triangle (C(n,r))

Calculate permutations (nPr, ordered arrangements) and combinations (nCr, unordered selections) of items, including versions that allow repetition or arrange items in a circle.

How It Works

How Permutation & Combination Calculator Works

In standard mode, permutations P(n,r) = n! / (n−r)! count the number of ordered arrangements of r items chosen from a set of n, while combinations C(n,r) = n! / (r!(n−r)!) count the same selections but ignore order — dividing out the r! different ways those r chosen items could themselves be arranged.

"With repetition" mode changes the counting rules entirely: permutations with repetition become n^r, since each of the r positions can independently be filled by any of the n available items (repeats allowed), while combinations with repetition use the "stars and bars" formula C(n+r−1, r).

Circular permutation mode computes (n−1)! instead of the usual n!, because rotating every item in a circular arrangement one position over doesn't create a genuinely new arrangement — fixing one item's position removes those redundant rotations from the count.

Worked Example

See It In Action

Choosing 3 items from a group of 10 (n=10, r=3): permutations P(10,3) = 10!/(10−3)! = 10×9×8 = 720 distinct ordered arrangements. Combinations C(10,3) = 720 / 3! = 720/6 = 120 distinct unordered selections — six times fewer, since each group of 3 chosen items could itself be arranged in 3! = 6 different orders.

Real-World Use Cases

Who Uses Permutation & Combination Calculator and Why

Figuring out how many distinct ways to award 1st, 2nd, and 3rd place among a group of competitors (a permutation problem).
Determining how many possible committees of a given size can be formed from a larger group (a combination problem).
Calculating how many PIN codes or passwords are possible when repeated digits or characters are allowed.
Working out how many distinct seatings are possible around a circular table, where rotations don't count as new arrangements.

Common Mistakes

Mistakes to Avoid

Using the permutation formula when order genuinely doesn't matter for the scenario, which overcounts the true number of distinct outcomes.
Forgetting that 'with repetition' fundamentally changes the underlying formula (n^r for permutations, or the stars-and-bars C(n+r−1, r) for combinations) rather than just being a minor tweak to the standard one.
Using n! instead of (n−1)! for a circular arrangement, which fails to account for rotations of the same arrangement being counted as identical.

Pro Tips

Tips for Best Results

Ask whether swapping two of your selected items creates a genuinely new outcome — if yes, you need a permutation; if no, you need a combination.
For circular arrangements, remember one item's position is effectively fixed to remove redundant rotations, which is why the formula uses (n−1)! instead of n!.

Troubleshooting

Fixing Common Problems

My result seems far larger or smaller than the real-world scenario suggests. — Double-check whether repetition should actually be allowed, and whether order genuinely matters for your scenario — both settings change the formula used and can produce very different magnitudes.

Glossary

Terms Explained

Factorial: The product of every positive integer up to a given number (n!), the building block for both permutation and combination formulas.

Stars and bars: A counting method behind the combinations-with-repetition formula C(n+r−1, r), used when the same item can be selected more than once.

FAQ

Frequently Asked Questions

What's the real-world difference between a permutation and a combination?

Use a permutation when order matters, like assigning 1st, 2nd, and 3rd place in a race. Use a combination when order doesn't matter, like choosing which 3 people join a committee — the same three people are the same committee no matter what order they were picked in.

How does "with repetition" change the calculation?

Without repetition, once an item is chosen it can't be chosen again. With repetition allowed, the same item can be selected multiple times — like choosing digits for a PIN code, where any digit can repeat in any position.

Why does circular permutation use (n−1)! instead of n!?

Arranging n items around a circle is considered the "same" arrangement no matter which item you start counting from, so the calculator fixes one item's position and only counts the (n−1)! ways to arrange everyone else relative to it.

What is Pascal's Triangle and how does it relate to combinations?

Each entry in Pascal's Triangle at row n, position r is exactly the value of C(n,r) — it's built by starting each row with 1 and summing the two numbers diagonally above to get every entry in between.