How to use the Combination Calculator
Enter n, the total number of items, and r, how many you choose. The result is the number of different groups you can make when the order does not matter.
Worked examples
Picking 2 toppings from 5 gives 5C2 = 10 possible pairs. A 5 card poker hand from a 52 card deck has 52C5 = 2,598,960 possibilities. A lottery that draws 6 numbers from 49 has 13,983,816 combinations, which is why a single ticket has about a 1 in 14 million chance.
Combination or permutation?
Ask whether swapping the order creates a different result. Choosing a committee of three people is a combination, because the same three people form the same committee. Awarding gold, silver and bronze is a permutation, because the order matters. For the same n and r there are always fewer combinations than permutations.
The formula
nCr = n! ÷ (r! × (n − r)!)
Examples
- C(52, 5) poker hands
C(52, 5) = 2,598,960
- C(10, 3)
C(10, 3) = 120
- C(49, 6) lottery
C(49, 6) = 13,983,816
Limitations and assumptions
Exact integer arithmetic.
Frequently asked questions
What is the combination formula?
nCr = n! ÷ (r! × (n − r)!).
Combination or permutation?
Use a combination when order does not matter, a permutation when it does.
What does nCr mean?
The number of ways to choose r items from n.
Why is nC0 equal to 1?
There is exactly one way to choose nothing.