What Are Permutations and Combinations?
Permutations and combinations are two sides of the same counting coin, and together they answer one of the most common questions in mathematics: how many ways can a set of things be arranged or chosen? The permutation and combination calculator on this page handles both instantly. Give it a total number of items and a number of items to pick, and it returns every ordered arrangement (a permutation) and every unordered group (a combination) that the selection allows. The two concepts look alike at a glance, but the single difference between them — whether order counts — completely changes the answer.
Imagine a shelf holding five books. Lining up any three of them produces an arrangement where the sequence matters, because the book placed first is different from the one placed second. Choosing three books to carry in a bag, by contrast, cares nothing about sequence, because the bag holds the same three books no matter how you drop them in. The first situation is a permutation; the second is a combination. This calculator lets you explore both on the same screen so the distinction becomes second nature.
When Order Matters Versus When It Does Not
The fastest way to decide which formula applies is to ask a single question: would swapping two chosen items change the outcome? If yes, you are counting permutations. If no, you are counting combinations. A ranking of three competitors treats first, second and third as three distinct outcomes, so it is a permutation problem. A committee of three people treats the same trio as one outcome regardless of who was named first, so it is a combination problem.
Real decisions are full of these distinctions. Assigning three different tasks to three people is a permutation because each task is unique and the pairing matters. Handing three identical tasks to three people is a combination because the group of people, not the assignment order, is what counts. Being able to see the two numbers side by side — nPr for the assignment and nCr for the group — makes the reasoning behind every counting question concrete rather than abstract.
The Factorial Foundation
Both formulas rest on the factorial, which mathematicians write with an exclamation point. The factorial of a whole number n is the product of n and every smaller positive whole number down to one. Five factorial is 5 × 4 × 3 × 2 × 1, which equals 120. Four factorial is 4 × 3 × 2 × 1, or 24. The factorial grows astonishingly quickly: ten factorial is already 3,628,800, and twenty factorial is more than two quadrillion.
Two conventions matter here. Zero factorial is defined to be one, rather than zero, and that choice is what makes the combination formula behave sensibly when you choose no items at all. The second convention is that factorials only apply to whole numbers, because the idea of multiplying down to one by every step loses its meaning for fractions. The calculator exposes the factorial of n directly as an output, labelled n!, so you can see how rapidly the total possible arrangements of the whole set expand as n grows.
How the Permutation Formula Works
The number of permutations of n items taken r at a time, written nPr, is computed as the factorial of n divided by the factorial of the difference between n and r:
nPr = n! ÷ (n − r)!
There is a natural way to think about why this works. To arrange r items in order, you have n choices for the first position, then (n − 1) for the second, then (n − 2) for the third, and so on until you have filled r positions. That gives the product n × (n − 1) × (n − 2) × … down to (n − r + 1). The factorial division simply cancels out everything below that point, leaving exactly the product you need. For ten items taken three at a time, that product is 10 × 9 × 8, or 720.
The permutation formula is the right tool whenever you are assigning ranks, ordering objects, lining people up, or building sequences where every position carries meaning. It appears constantly in problems about seating arrangements, race finishes, passwords, and scheduling, and it is the reason a modest set of ten items can still produce more than seven hundred distinct three-item arrangements.
How the Combination Formula Works
The number of combinations of n items taken r at a time, written nCr, removes the ordering that permutations count. The formula divides the permutation result by the factorial of r:
nCr = n! ÷ (r! × (n − r)!)
Every group of r items can be written out in r! different orders, and since a combination ignores order, all of those internal arrangements collapse into a single outcome. Dividing by r! removes exactly that redundancy. For ten items taken three at a time, the permutation count of 720 is divided by 3!, which is 6, producing 120 distinct groups. That is the answer to the classic question of how many three-person committees can be chosen from ten people.
Because a single group can be ordered in several ways, nCr is always smaller than nPr for the same inputs, and the gap grows with r. The two numbers together tell a complete story: nPr counts the ordered outcomes, nCr counts the unordered ones, and the ratio between them is exactly r!. Seeing both results at once is the clearest way to appreciate what order is really worth in a counting problem.
Choosing Your Inputs
Two controls drive the calculator, and both are simple:
- Number of items (n) — the total size of the set you are choosing from, from a modest 2 up to 100.
- Items chosen (r) — how many items you pick from that set, from 0 up to 100.
There is no restriction preventing you from setting r above n, but it is important to know what happens when you do. Selecting more items than the set contains is impossible, so the results return zero rather than attempting a calculation. You can slide r back below n at any time and the numbers immediately return. For most problems you will keep r comfortably smaller than n, often far smaller, since choosing more than half the set starts to shrink the available combinations.
Reading the Results
The results panel reports four numbers:
- Permutations (nPr) — every ordered arrangement of r items drawn from n.
- Combinations (nCr) — every unordered group of r items drawn from n.
- Total arrangements (n!) — every possible ordering of all n items at once.
- Combinations as share of n! — how much of the full set of arrangements your groups represent, shown as a percentage.
The bar chart underneath plots permutations and combinations together as r moves from zero toward n. The two curves tell the same story the numbers do: permutations tower above combinations, both climb to a peak as r approaches the middle of the range, and both collapse back toward one at the extremes. It is a compact visual summary of everything the formulas describe.
A Worked Example
Suppose you are organising a quiz night with ten contestants and you need to award prizes for first, second and third place. Since the podium positions are distinct, this is a permutation: 10P3 equals 720, so there are 720 possible podium line-ups. Now suppose instead you need to pick a three-person team from the same ten contestants, with no captain and no ranking. This is a combination: 10C3 equals 120, so there are 120 possible teams.
Notice how the three podium positions produce six times as many outcomes as the three-person team. That factor of six is exactly 3!, the number of ways the three team members can be arranged onto the podium. The calculator lets you verify this relationship by eye: set n to 10 and r to 3, and you will see the permutations at 720 and the combinations at 120, with n! standing at over 3.6 million to remind you how large the full set of arrangements really is.
Common Real-World Applications
The same two formulas appear across a remarkable range of everyday situations. In probability, the chance of a particular lottery draw is one divided by the number of combinations the draw allows, which is why 6-from-49 lotteries use the combination formula. In sports and games, the number of possible fixtures, pairings, and starting line-ups all reduce to nCr. In computer science, counting the ways to select or order data items underpins many algorithms, and in business, everything from forming project teams to arranging a seating chart is a counting problem in disguise.
Because the distinction between ordered and unordered is so fundamental, mastering nPr and nCr transfers directly to more advanced topics such as binomial coefficients and the binomial theorem. The coefficients of a binomial expansion are in fact the combination values nCr, which is why the calculator is useful well beyond simple counting drills. Whatever your level, playing with the sliders and watching the chart respond is one of the fastest ways to build an intuition for how rapidly the possible outcomes multiply.
Tips and Traps
A few cautions will save you from common mistakes. First, always confirm that order truly does not matter before reaching for the combination formula; using the wrong one changes the answer dramatically. Second, remember that these formulas assume no repetition, so they do not apply to situations where an item can be reused. Third, keep an eye on r: choosing exactly r equal to n gives a single combination (you take everything) and nPr equal to n!, while choosing r equal to zero gives a single empty selection by the convention that 0! is one.
Finally, be aware of scale. Factorials grow faster than almost any function you meet in school mathematics, so once n rises past twenty or thirty, the numbers become astronomically large and begin to lose exact precision on a computer. The calculator still returns a useful figure, but for those enormous ranges it is the general magnitude that matters rather than the final digits. For the everyday counting problems most people encounter, the results are exactly right.
Related Calculators
Counting and statistics go hand in hand, so several nearby tools can help you dig deeper. The factor calculator explores divisibility, while the math calculator offers a general-purpose tool for everyday arithmetic. If you are combining counting with data analysis, the statistics calculator, percentage calculator and average calculator round out a useful toolkit for problem sets, exams and project work alike.
Disclaimer
Results are provided as estimates for informational purposes only and may be inaccurate. Always verify outcomes with a qualified professional before making financial or personal decisions based on these calculations.