Combination Calculator
Solve combination problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Combination Calculator
Calculator
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Formula: C(n,r) = n! / (r!(n-r)!)
Worked example — C(12,4) = 495 possible committees
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.
What real-world problems use combinations?
Combinations are used to calculate lottery odds (choosing 6 numbers from 49 with order irrelevant), poker hand probabilities (choosing 5 cards from a 52-card deck), committee formation (selecting 4 people from a department of 15 with no distinct roles), and quality-control sampling (choosing which 10 units from a batch of 200 to inspect).
How large can n get before factorials become impractical to compute directly?
Factorials grow extremely fast — 20! is already over 2.4 quintillion, and 170! exceeds the range of standard double-precision floating point. In practice, combination calculators avoid computing the full factorials separately and instead cancel common terms in the numerator and denominator (or use logarithms) to keep intermediate values manageable even for large n and r.
How is C(n,r) connected to Pascal's Triangle and the binomial theorem?
Every entry in Pascal's Triangle is a combination value — the r-th entry (from 0) in row n equals C(n,r). These same values are the coefficients in the binomial expansion of (x+y)ⁿ, which is why C(n,r) is also called a binomial coefficient. This connects combinatorics directly to algebra: expanding (x+y)⁵ produces the coefficients 1, 5, 10, 10, 5, 1 — exactly row 5 of Pascal's Triangle.
How do combinations and permutations differ?
Combinations count selections where order does not matter — choosing 5 cards from 52 is a combination (C(52,5) = 2,598,960). Permutations count arrangements where order matters — arranging 5 cards in a specific sequence is a permutation. Lottery jackpots use combinations; ranking finishers in a race uses permutations.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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