Combination Calculator
Find how many unique groups you can form when selecting r items from a set of n, where order doesn't matter. Perfect for lottery odds, team selection, and probability problems.
About this calculator
A combination counts the number of ways to choose r items from n distinct items when the order of selection is irrelevant. The formula is C(n, r) = n! / (r! × (n − r)!), where '!' denotes factorial — the product of all positive integers up to that number. For example, choosing 3 items from 5 gives C(5,3) = 120 / (6 × 2) = 10 unique groups. Combinations differ from permutations in that {A, B, C} and {C, B, A} count as the same group. This concept is foundational in probability, statistics, and combinatorics, used whenever you need to count distinct subsets — such as lottery number selections, committee formations, or hand combinations in card games.
How to use
Suppose you want to know how many 3-person committees can be formed from a group of 8 people. Enter n = 8 (total people) and r = 3 (committee size). The calculator computes: C(8, 3) = 8! / (3! × 5!) = 40320 / (6 × 120) = 40320 / 720 = 56. So there are 56 unique committees possible. Try n = 52 and r = 5 to find the number of possible 5-card poker hands: 2,598,960.
Frequently asked questions
What is the difference between combinations and permutations?
Combinations count groupings where order does not matter, while permutations count arrangements where order does matter. For example, choosing team members {Alice, Bob} is the same combination regardless of who is picked first, but in a permutation, (Alice, Bob) and (Bob, Alice) are counted separately. The combination formula is C(n,r) = n! / (r! × (n−r)!), whereas the permutation formula is P(n,r) = n! / (n−r)!. Use combinations when you only care about which items are selected, not the sequence.
How do you calculate combinations when n and r are large numbers?
When n and r are large, factorials grow astronomically fast, making direct computation impractical by hand. You can simplify by cancelling terms — for example, C(10,3) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120, without computing 10! in full. Most calculators and programming languages handle this automatically using optimized algorithms. It's also worth noting that C(n,r) always equals C(n, n−r), so you can use whichever pair involves the smaller factorial for easier manual computation.
When should I use a combination calculator in real life?
Combination calculators are useful any time you need to count distinct selections without regard to order. Common real-world uses include calculating lottery odds (e.g., picking 6 numbers from 49), determining how many ways a sports bracket can be filled, counting possible card hands in poker, or figuring out how many research sample groups can be drawn from a population. Anytime a problem asks 'how many ways can you choose X from Y,' a combination calculation is the right approach.