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Permutation and Combination Calculator: nPr, nCr and n!

This free calculator finds permutations (nPr), combinations (nCr) and factorials (n!) exactly, using big integers so the digits are never rounded. It shows the formula it used.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

Permutations and combinations count the ways to pick and arrange items. They are the base of probability and appear in passwords, lotteries, team selection and card games. The numbers grow very fast, so a normal calculator quickly shows only an approximation.

This tool uses exact integer arithmetic, so even 52C5 or 100! is shown digit for digit.

How to Use This Tool

  1. Enter n, the total number of items.
  2. Enter r, how many you choose (skip it for factorial).
  3. Pick what to calculate.
  4. Press Calculate and read the exact value and the formula.

How the Calculation Works

The factorial of n multiplies every whole number from 1 up to n. A permutation is nPr = n × (n − 1) × … × (n − r + 1). A combination divides that by r!, because each group of r items can be ordered in r! ways that count as one choice.

The tool multiplies and divides whole numbers only, so there is no rounding. Very long results are shown as the full number plus the digit count.

Worked Example

How many ways can 3 medals be given among 10 runners? Order matters, so use 10P3 = 10 × 9 × 8 = 720. How many ways can you pick a team of 3 from the same 10? Order does not matter, so 10C3 = 720 ÷ 6 = 120.

Common Uses

  • Probability homework and exam practice.
  • Counting password or PIN possibilities.
  • Lottery odds, for example 6 numbers from 49 gives 13,983,816 combinations.
  • Planning seating or tournament schedules.

Tips and Common Mistakes

  • r cannot be bigger than n.
  • nCr is the same as nC(n − r), which is a quick check.
  • If items can repeat, the formulas are different. This tool covers selection without repeats.

Next, you may need the Fibonacci calculator, GCD and LCM calculator, prime number checker, average calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

A permutation counts arrangements where the order matters, such as a race podium. A combination counts selections where order does not matter, such as a lottery ticket.
nPr = n! ÷ (n − r)!, and nCr = n! ÷ (r! × (n − r)!). A factorial n! is the product of all whole numbers from 1 to n.
By definition 0! = 1. It keeps the formulas for nPr and nCr working when r is 0 or equal to n.
Up to 5,000 for factorials and up to 100,000 for nPr and nCr. Results with hundreds of digits are shown in full.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.