Combinations & Permutations Calculator
Compute combinations (nCr) and permutations (nPr), with and without repetition, plus factorials — the core counting formulas for probability and statistics. Free, runs in your browser.
Combinations & Permutations Calculator
Whole numbers. Order-matters gives permutations; order-doesn't gives combinations — each with and without repetition.
How to Use the Combinations & Permutations Calculator
Enter n and r
Type the total number of items n and how many you are choosing r — for example 10 items, choosing 3.
Read all four counts
The calculator shows combinations and permutations, each both without repetition (no item reused) and with repetition (items may repeat).
Pick the right one
Use permutations when order matters (a podium, a PIN) and combinations when it does not (a committee, a lottery draw). The labels say which is which.
See the factorials
The factorial values n!, r! and (n − r)! that the formulas are built from are shown too, so you can follow the working.
Counting Without Listing
Order Matters, or It Doesn't
Combinatorics is the art of counting possibilities without writing them all out, and almost every question it answers comes down to a single distinction: does order matter? If you are arranging three runners on a podium, gold–silver–bronze is different from bronze–silver–gold, so order matters and you are counting permutations. If you are choosing three pizza toppings, the set is the same however you list it, so order does not matter and you are counting combinations. These two ideas — permutations (nPr) and combinations (nCr) — are the backbone of probability, statistics and discrete mathematics, and this calculator computes both at once from your n and r, alongside the two variants where repetition is allowed. The numbers grow astonishingly fast: there are only 10 ways to choose 3 items from 5, but over 2.5 million ways to deal a five-card poker hand from 52, and the number of ways to order a single deck of cards exceeds the number of atoms in our galaxy many times over.
The engine behind all of this is the factorial — n!, the product of every whole number from 1 up to n — which counts the ways to arrange n distinct items in a row. Permutations and combinations are both ratios of factorials: nPr divides n! by (n − r)! to count ordered selections, and nCr divides further by r! because, for combinations, the r! different orderings of the same chosen set should be counted only once. That extra division is the whole difference between the two, and it is why nCr is always the smaller number. When repetition is allowed the formulas shift: ordered selections with repeats are simply n raised to the power r, while unordered selections with repeats use the elegant "stars and bars" formula. This tool shows all four, plus the underlying factorials, so the relationships are visible rather than hidden. Because the values overflow ordinary precision quickly, the combination and permutation counts are computed step by step to stay exact for as long as possible, and very large factorials are flagged rather than shown as a meaningless rounded figure. Everything runs in your browser.
"Permutations count arrangements; combinations count selections. The only difference is whether order matters — and that single question decides which formula you need."
Combinatorics in Everyday Life
These counts are everywhere a probability is calculated. The odds of a lottery are one divided by the number of combinations of balls; the chance of a particular poker hand is its count divided by the total number of hands. Password strength is a permutation-with-repetition count — the number of possible strings of a given length — and it is why each extra character multiplies security. In statistics, the binomial coefficient nCr appears in the binomial distribution, in Pascal's triangle, and in the expansion of (a + b) to a power. Scheduling, seating plans, tournament brackets, genetics, and the analysis of algorithms all reduce to counting arrangements or selections. Even the birthday paradox — that just 23 people give a better-than-even chance of a shared birthday — is a combinatorial calculation that surprises almost everyone. Whether you are working out a probability, checking statistics homework, estimating how many possibilities a system has, or simply curious how fast the numbers grow, this calculator gives you every standard count instantly and privately.
10 Facts About Counting
Permutations count arrangements where order matters.
Combinations count selections where order does not matter.
nCr is always smaller than nPr — it divides out the orderings.
A factorial n! counts the ways to arrange n distinct items.
There are 2,598,960 possible five-card poker hands.
Ordering one deck of cards has 52! arrangements — astronomically many.
The binomial coefficient nCr fills Pascal's triangle.
Password strength is a permutation-with-repetition count.
Just 23 people give a 50% chance of a shared birthday.
This calculator runs in your browser — nothing is uploaded.
Frequently Asked Questions
- A permutation counts selections where order matters — gold, silver and bronze on a podium are a different result if you swap them. A combination counts selections where order does not matter — a committee of three people is the same group however you list them. Permutations are always at least as many as combinations.
- Without repetition, each item can be chosen only once — like dealing cards. With repetition, an item can be picked again — like choosing scoops of ice cream where you can repeat a flavour, or digits in a PIN. The calculator shows both cases for combinations and permutations.
- nCr equals n! divided by r! times (n − r)!. It takes the number of ordered selections, nPr, and divides by r! because each unordered combination corresponds to r! different orderings that should be counted only once. The calculator computes it step by step to stay exact.
- The factorial of n, written n!, is the product of all whole numbers from 1 up to n — so 5! = 1 × 2 × 3 × 4 × 5 = 120. It counts the number of ways to arrange n distinct items in order, and it is the building block of both the permutation and combination formulas.
- Ask whether order matters. For a lottery draw, a committee, or a hand of cards, order does not matter — use combinations. For a ranking, a podium, a PIN, or an arrangement in a row, order matters — use permutations. The calculator labels each result so you can pick the right one.
- Factorials and counts grow extremely fast — beyond a certain size they exceed the exact range of ordinary numbers. Rather than show a rounded, misleading figure, the calculator flags such values as too large to display while still computing the combination and permutation counts exactly for as long as it can.
- For selections without repetition, no — you cannot choose more distinct items than exist, so those results are zero. But the with-repetition counts still make sense when r exceeds n, since items can be reused, and the calculator continues to show those.
- In probability and statistics constantly: lottery and card odds, the binomial distribution, Pascal's triangle, and password-strength estimates. They also appear in scheduling, tournament design, genetics and algorithm analysis — anywhere you need to count arrangements or selections.
- No. Every calculation runs in your browser with a small built-in engine — nothing is uploaded to a server or third-party library, and the tool works offline once the page has loaded.
- Completely free, with no account, sign-up, or usage limit. It runs entirely in your browser and collects no data. Use it as much as you like.
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