Combination Calculator

A combination counts the ways to choose a group from a larger set when the order of choosing does not matter — committees, lottery numbers, card hands, tasting panels, samples. The count grows far faster than intuition suggests: choosing just 5 from 49 already produces nearly two million groups. This combination calculator applies the standard n-choose-r formula to your numbers, returning the exact count so that lottery odds, survey samples and team selections rest on arithmetic rather than guesswork.

Quick Answer

C(n, r) = n! ÷ (r! × (n − r)!) — read "n choose r." Divide the factorial of the total pool by the product of the factorials of the chosen count and the unchosen count. For example, C(10, 3) = 120.

What the Formula Is Really Doing

Imagine lining up all n items and picking r of them one at a time: n choices, then n−1, then n−2, giving n!/(n−r)! ordered picks. But a selection does not care about the order its members were picked in, and each group of r items can be picked in r! different orders. Dividing the ordered count by r! removes that overcounting and leaves the number of distinct groups — that division is the whole logic of the formula, and it is why combinations are always smaller than the matching permutations.

Two elegant consequences follow. First, symmetry: C(n, r) = C(n, n−r), because choosing which r items to take is the same act as choosing which n−r items to leave behind — C(20, 17) equals C(20, 3) = 1,140, since picking 17 to include means picking 3 to exclude. Second, the edges: C(n, 0) = C(n, n) = 1, because there is exactly one way to choose nothing and one way to choose everything. Factorial, written n!, means n × (n−1) × ... × 1, and the formula assumes every item is distinct and chosen at most once.

How to Use the Combination Calculator

1. Count the total number of distinct items available — that is n.
2. Decide how many will be in each group — that is r.
3. Confirm order genuinely does not matter in your situation; if rankings or positions are involved, this is the wrong formula.
4. Enter n and r into the calculator's fields.
5. Read the number of possible combinations.
6. Use the symmetry shortcut when r is more than half of n: compute C(n, n−r) instead — the result is identical and the arithmetic is lighter to check by hand.

Worked Example: Choosing 3 Finalists from 10

A quiz league must send a three-person team from a squad of 10, and every member of the team has the same role. Count the possible teams: C(10, 3) = 10! / (3! × 7!). Cancelling the 7! leaves (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120 possible teams. The cancellation step is the practical way to evaluate any combination by hand: write r descending factors on top, r! below, and divide. For scale, notice what happens when the numbers grow: a lottery that draws 5 numbers from 49 offers C(49, 5) = 1,906,884 possible lines, which is why a single ticket's chance of matching all five is one in roughly 1.9 million — the combination count is the denominator of that probability, computed before any question of luck.

Benchmark values:

• C(5, 2) = 10
• C(6, 3) = 20
• C(12, 4) = 495
• C(20, 3) = 1,140 (and therefore C(20, 17) = 1,140)
• C(52, 5) = 2,598,960 (the number of distinct five-card poker hands)

When the Combination Formula Gives the Wrong Answer

The formula is exact, but three situations quietly fall outside it. First, ordered outcomes: if the group being formed has internal roles — a captain and vice-captain, a first and second prize — the count is a permutation, and using combinations understates it by a factor of r!. Second, repetition: if items can be picked more than once (choosing scoops where the same flavour may repeat), a different formula applies and this one undercounts. Third, non-distinct items: the formula assumes every item in the pool is distinguishable, so pools with identical members need adjusted methods. A practical caution completes the list: combination counts describe how many outcomes exist, not which one will occur. Knowing there are 1,906,884 lottery lines explains the odds honestly — it does not make any line likelier, and no pattern of "due" numbers changes the count.

Sampling Without Opening Every Box

Quality control runs on combinations. A warehouse receives a batch of 50 units and inspectors will open and test just 4 of them, chosen at random. The number of different samples they might draw is C(50, 4) = (50 × 49 × 48 × 47) / 24 = 230,300 — nearly a quarter of a million possible samples from a modest batch and a small sample size. That count is what makes random sampling defensible: with so many equally likely samples, a defect spread through the batch has a calculable chance of appearing in whichever sample is drawn, and the inspection plan can be justified with arithmetic instead of optimism. The same structure sizes clinical trial groups, audit selections and acceptance tests. It also explains why "we checked a few and they were fine" is a weak claim when the few were chosen by convenience rather than at random — the formula counts what random choice can reach, and convenience samples never leave the first shelf of that space. When the batch is defective at a known rate, the counts from this calculator feed directly into the probability of catching at least one bad unit.

Counting, Then Comparing Chances

Selections are only half the story — most people count combinations in order to weigh a chance, and the Probability Calculator picks up exactly there, turning counts of favourable and total outcomes into probabilities. If your situation mixes arrangements and selections, the Permutation and Combination calculator sets the two side by side so the order question can be settled before either number is trusted. Both are part of the Math tools, alongside the factorial, statistics and algebra calculators that counting problems tend to pull in next.