Permutation & Combination Calculator — Free Online Tool

What Is Permutation & Combination Calculator?

A free online permutation and combination calculator. Compute nPr (permutations where order matters) and nCr (combinations where order does not matter) for any n and r values. Includes factorial breakdown for full transparency.

Permutations and combinations are fundamental counting techniques in probability, statistics, and combinatorics. The key difference: permutations (nPr) count arrangements where order matters — like choosing 1st, 2nd, and 3rd place winners from 10 contestants. Combinations (nCr) count selections where order does not matter — like choosing 3 committee members from 10 people. Permutation & Combination Calculator computes both with factorial breakdowns.

Enter the total number of items (n) and the number of items to choose (r). The tool instantly displays both the permutation (nPr) and combination (nCr) values. The formula view shows the full calculation: nPr = n! / (n-r)! and nCr = n! / (r! × (n-r)!).

The factorial breakdown shows the step-by-step computation: n!, r!, (n-r)!, and the final division. This transparency helps students understand where the numbers come from and verify the calculation manually. For n=10, r=3: 10! = 3,628,800, 7! = 5,040, so 10P3 = 3,628,800/5,040 = 720, and 10C3 = 3,628,800/(6 × 5,040) = 120.

The tool handles large values efficiently using optimized factorial computation. For very large n values (over 170), standard JavaScript numbers overflow, so the tool displays "Infinity" for results exceeding Number.MAX_VALUE. Results update instantly as you adjust n and r values.

Real-World Use Cases for Permutation & Combination Calculator

Counting Possible Rankings in a Competition

10 contestants compete for gold, silver, and bronze medals. Order matters (1st, 2nd, 3rd are distinct). Use n=10, r=3: 10P3 = 720 possible podium arrangements. The factorial breakdown shows 10!/7! = 720.

Choosing a Committee from a Group

You need a 3-person committee from 10 people. Order does not matter (being "on the committee" is the same regardless of selection order). Use n=10, r=3: 10C3 = 120 possible committees. Compare this with 720 permutations — combinations always give fewer results.

Calculating Lottery Odds

A lottery requires choosing 6 numbers from 49. Order does not matter (any matching set of 6 wins). Use n=49, r=6: 49C6 = 13,983,816 possible combinations. The odds of winning with one ticket are 1 in ~14 million.

Why Use Permutation & Combination Calculator?

Permutations (nPr) and combinations (nCr)
Factorial breakdown for n!, r!, and (n-r)!
Handles large values efficiently
Useful for probability and counting problems

How to Use Permutation & Combination Calculator — Step by Step

Enter the total number of items (n).

Enter the number of items to choose (r).

View nPr (permutations) and nCr (combinations) instantly.

Review the factorial breakdown for each component.

Who Is Permutation & Combination Calculator Best For?

probability problems
statistics
lottery odds
math competitions

Pro Tips for Permutation & Combination Calculator

1Remember the key question: does order matter? If yes, use permutations (nPr). If no, use combinations (nCr). Picking a president, VP, and treasurer is a permutation. Picking a committee is a combination.
2nPr is always larger than nCr for the same n and r (except when r=0 or r=1, where they are equal). This makes sense: ordered arrangements are more numerous than unordered selections.
3The factorial (!) of a number is the product of all positive integers up to that number. 5! = 5×4×3×2×1 = 120. 0! is defined as 1.
4When r = n, nPr = n! (all possible orderings) and nCr = 1 (only one way to choose everything). When r = 0, both nPr and nCr equal 1.

Permutation & Combination Calculator — Frequently Asked Questions

What is the difference between permutation and combination?

Permutations (nPr) count arrangements where order matters. Combinations (nCr) count selections where order does not matter. For example, picking 3 winners from 10 — if order of winners matters, use permutation.

Why does my result show NaN?

If r is larger than n, or if negative values are entered, the result will be NaN (not a number). Ensure n ≥ r and both are non-negative.

How accurate are the calculations?

All calculations use double-precision floating-point arithmetic following IEEE 754 standards. Financial calculations like loans and compound interest are verified against standard spreadsheet formulas.

Can I round results to a specific number of decimal places?

Results are typically displayed with up to 10 significant digits. You can mentally round to your desired precision for most practical use cases.

Is Permutation & Combination Calculator really free to use?

Yes, Permutation & Combination Calculator is completely free with no hidden charges, no usage limits, and no premium tiers. You can use it as often as you need without signing up or providing any personal information.

Ready to Use Permutation & Combination Calculator?

Free online tool — works in your browser, no sign-up required. Start using it right now.

Try It Now