What is a combination?
A selection where order does not matter. Choosing 2 pizza toppings from 5 is a combination, because cheese-then-mushroom is the same pizza as mushroom-then-cheese.
// maths › Probability
Combination Calculator
How many unordered selections of r items can be drawn from n, where order does not matter.
nCr = n! / (r! (n − r)!)
Frequently asked questions
Why divide by r! compared with permutations?
Because every group of r items can be arranged in r! different orders, and combinations should count all those as one. Dividing the permutation count by r! removes the duplicate orderings.
When do I use combinations versus permutations?
Ask yourself whether reordering the same items makes a different outcome. Lottery numbers, committee members and hands of cards are combinations; rankings, codes and finishing orders are permutations.
What is nCr also called?
The binomial coefficient, often read as 'n choose r'. It is the same number that appears in Pascal's triangle and the binomial expansion.
A quick real example?
Picking 6 lottery numbers from 45 is a combination because the draw order does not matter. The huge number of combinations is exactly why the jackpot is so hard to win.