Combinations & Permutations Calculator
Enter n and r to get exact nCr and nPr values instantly, with or without repetition, including the worked formulas.
Combinations without repetition (nCr)
120
C(n, r) = n! / (r! × (n − r)!)
C(10, 3) = 10! / (3! × 7!)
Permutations without repetition (nPr)
720
P(n, r) = n! / (n − r)!
P(10, 3) = 10! / 7!
Factorial of n (10!)
3,628,800
Exact BigInt arithmetic, computed locally in your browser. n and r up to 5,000.
How it works
- Enter n, the total number of items in your set, and r, the number of items you are choosing or arranging.
- Toggle Allow repetition if the same item may be picked more than once — the formulas switch to C(n+r−1, r) and n^r automatically.
- Read the exact nCr and nPr results with the substituted formulas below each one, and use the copy buttons to grab the raw numbers.
Frequently asked questions
- What is the difference between a combination and a permutation?
- A combination counts selections where order does not matter, such as choosing 3 pizza toppings from 10. A permutation counts arrangements where order matters, such as ranking 3 finalists out of 10 contestants. That is why nPr is always at least as large as nCr: it multiplies each combination by the r! ways to order it.
- Are the results exact for very large numbers?
- Yes. The calculator uses JavaScript BigInt arithmetic, so every digit is exact — there is no floating-point rounding, even for results with thousands of digits. When a result exceeds 15 digits, a scientific-notation approximation is shown alongside the full number for readability.
- What do the repetition formulas mean?
- With repetition allowed, combinations use the multiset formula C(n+r−1, r), which counts selections where an item can be picked more than once — like choosing 4 scoops from 6 ice-cream flavors. Permutations with repetition become n^r, since each of the r positions can independently take any of the n items, like 4-digit PIN codes from 10 digits.
- Why do I get 0 when r is larger than n?
- Without repetition you cannot choose or arrange more items than the set contains, so both C(n, r) and P(n, r) are defined as 0 when r exceeds n. If you turn on Allow repetition, larger r values become valid because items can be reused, and the results grow accordingly.
- Is anything I type sent to a server?
- No. All calculations run entirely in your browser using client-side BigInt math — nothing is uploaded, logged, or stored anywhere. You can even load the page and then disconnect from the internet; the calculator keeps working normally.
About this tool
This calculator computes combinations (nCr) and permutations (nPr) for any pair of whole numbers up to n = 5,000 and r = 5,000, updating live as you type. Both results appear at once, together with n! (n factorial) and the formulas rendered with your actual values substituted in, so you can see exactly how each answer was derived — useful for checking homework or verifying a step in a probability problem.
Everything runs client-side using JavaScript’s BigInt type, which performs integer arithmetic with unlimited precision. Ordinary floating-point math loses accuracy beyond about 15 digits, but BigInt keeps every digit exact, so C(1000, 500) is shown in full — all 300 digits of it — with thousands separators. For results longer than 15 digits a compact scientific approximation such as 2.703 × 10^299 is displayed next to the exact value. Because no data leaves your device, the tool is private by design and works offline once loaded.
The repetition toggle switches between the two classic counting models. Without repetition, combinations follow C(n, r) = n! / (r! × (n − r)!) and permutations follow P(n, r) = n! / (n − r)!. With repetition allowed, combinations use the multiset formula C(n+r−1, r) — the standard stars-and-bars result — and permutations become simply n^r. These cover most practical counting questions: lottery odds, poker hands, seating arrangements, password and PIN spaces, team selections, and sampling problems in statistics.
A practical tip: when r is more than half of n, remember that C(n, r) equals C(n, n−r), which is why choosing 48 items from 50 gives the same small answer as choosing 2. And if you only need an order-of-magnitude estimate for a huge permutation count, the scientific approximation shown beside each result is usually all you need — copy the exact value only when downstream software requires full precision.