Combination and permutation calculator
How many ways to choose r items from n: combinations (nCr), permutations (nPr), with or without repetition, plus n factorial.
Does order matter? See every arrangement
Choose 2 letters from ABCD. Toggle order on and AB and BA become different outcomes; the count grows by a factor of 2! = 2.
- AB
- AC
- AD
- BC
- BD
- CD
Show me the steps
An AI tutor walks through the working for any problem you type. The answer above is computed in your browser; the explanation is optional and is the only thing that leaves your device.
Questions people ask
What is the difference between a combination and a permutation?
A permutation is an ordered arrangement; a combination is an unordered selection. Picking a president, secretary and treasurer from 10 people is a permutation (10P3 = 720). Picking a 3-person committee is a combination (10C3 = 120), because the same three people in a different order are the same committee.
What are the formulas?
nPr = n! / (n − r)! and nCr = n! / (r!(n − r)!). nCr is nPr divided by r!, the number of ways to order each selection.
What are the odds of winning a 6/49 lottery?
Order does not matter, so it is 49C6 = 13,983,816 possible tickets: about 1 in 14 million for the jackpot.
What is a factorial?
n! is the product of all whole numbers from 1 to n: 5! = 120. By definition 0! = 1. Factorials grow very fast; 171! exceeds the largest number a double-precision float can hold.
When do I use “with repetition”?
When the same item can be chosen more than once: a 4-digit PIN from 10 digits is 10⁴ = 10,000 (permutations with repetition). Choosing 3 scoops from 5 ice-cream flavours, repeats allowed, is C(5+3−1, 3) = 35.