Combination Calculator
Solve combination problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
C(n,r) = n! / (r!(n-r)!)
Combinations count the number of unordered subsets of size r that can be selected from a set of n distinct items. It equals the number of ordered arrangements P(n,r) = n!/(n-r)! divided by r! to remove duplicate orderings of the same group.
Worked Examples
Example 1: Choosing a committee
Problem:How many ways can a 4-person committee be formed from a group of 12 people?
Solution:C(12,4) = 12! / (4! × 8!) = (12×11×10×9) / (4×3×2×1) = 11,880 / 24 = 495.
Result:C(12,4) = 495 possible committees
Example 2: Poker hand probability setup
Problem:How many different 5-card hands can be dealt from a standard 52-card deck?
Solution:C(52,5) = 52! / (5! × 47!) = (52×51×50×49×48) / 120 = 311,875,200 / 120 = 2,598,960.
Result:C(52,5) = 2,598,960 possible hands
Frequently Asked Questions
What is a combination in mathematics?
A combination is a selection of r items from a larger set of n items where the order of selection does not matter. C(n,r), read 'n choose r', counts how many distinct unordered subsets of size r can be formed from n items. Choosing {A, B, C} is the same combination as choosing {C, B, A} — the group is identical either way.
What is the difference between a combination and a permutation?
A combination ignores order (choosing 3 pizza toppings from 10 — the order you name them doesn't create a different topping set), while a permutation counts order as significant (assigning 1st, 2nd, and 3rd place medals from 10 racers — the same 3 people in a different order is a different outcome). Because permutations count every ordering separately, P(n,r) is always C(n,r) × r! — larger than the corresponding combination whenever r > 1.
How is the combination formula derived?
Start with the number of ordered arrangements of r items from n, which is P(n,r) = n!/(n-r)!. Since each unordered group of r items can be arranged in r! different orders, dividing by r! removes the duplicate orderings: C(n,r) = n! / (r! × (n-r)!). This division is exactly why the combination count is always smaller than or equal to the permutation count.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy