Combination formula
A combination counts the ways to choose r items from n when order does not matter. Selecting the same group in a different order counts only once, so combinations give smaller numbers than permutations.
- n = the total number of items
- r = the number of items chosen
- n! = n factorial
When order does not matter
Use combinations when only the group matters, not its arrangement, such as choosing a committee, dealing a hand of cards, or picking lottery numbers. The extra r factorial in the denominator divides out the duplicate orderings that permutations would count separately.
- Choose 2 items from 5, so n = 5 and r = 2.
- Apply the formula: C(5, 2) = 5! / (2! × 3!).
- That is 120 / (2 × 6) = 120 / 12 = 10.
- There are 10 unordered selections.
Related formula
See the Permutation Formula for arrangements where order matters.
FAQ
What is the combination formula?
C(n, r) = n! / (r!(n − r)!), the number of ways to choose r items from n when order does not matter.
Why is a combination smaller than a permutation?
Combinations ignore order, so each unordered group corresponds to several ordered permutations. Dividing by r! removes those duplicates.