Permutation and combination calculator
Enter n items and r chosen. You get the number of ordered arrangements (permutations) and unordered selections (combinations), computed exactly and shown in full or in scientific notation when long.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Choose without or with repetition.
- Enter n, the number of items to pick from, and r, the number you pick.
- Both counts appear at once, with the formula that produced them.
How it works
Without repetition, nPr is the product n × (n−1) × … × (n−r+1). nCr is built up one factor at a time, each step an exact whole-number division, using the smaller of r and n−r to keep it short. With repetition the tool uses n^r and C(n+r−1, r).
Worked example
Five cards from a standard 52-card deck:
- Ordered hands: 52 × 51 × 50 × 49 × 48 = 311,875,200.
- Unordered hands: 311,875,200 / 5! = 311,875,200 / 120 = 2,598,960.
With repetition, a 3-digit code from the digits 0–9 has 10³ = 1,000 possibilities.
Tips
- Ask yourself whether swapping two chosen items gives a different outcome. If yes, use permutations.
- A lock “combination” is really a permutation, because order matters.
- The numbers grow fast: C(100, 50) already has 30 digits.
Frequently asked questions
What is the difference between a permutation and a combination?
In a permutation the order matters, so picking A then B differs from B then A. In a combination only the chosen set matters. Choosing 3 people from 10 for a ranked podium has 720 outcomes (permutations); choosing 3 for a committee has 120 (combinations).
What are the formulas for nPr and nCr?
nPr = n! / (n − r)! and nCr = n! / (r! (n − r)!). Every combination can be ordered in r! ways, which is why nCr = nPr / r!.
What changes when repetition is allowed?
Ordered choices with repetition number n^r, since each of the r positions can take any of the n items. Unordered choices with repetition, picking scoops from n flavors, number C(n + r − 1, r).
How are huge results handled?
The tool uses arbitrary-precision integers, so there is no rounding. Results over 24 digits are shown in scientific notation first and the exact digits follow. Copy gives every digit.
What do 0 and r = n give?
Choosing nothing or choosing everything can be done in exactly one way, so nCr is 1 for r = 0 and for r = n. nPn equals n!, and nP0 equals 1.