Combination Calculator
Why people count combinations
How many five-card poker hands can exist? Count, do not list: C(52, 5) = 2,598,960. The Combination Calculator answers exactly this style of question, selecting k items from n when order does not matter. Quiz setters, lottery players, logistics planners, and probability students all call on it, and the tool works without repetition by default.
The combination formula in action
Combinations use C(n, k) = n! / (k! (n-k)!), which cancels the ordering overlap. Small case: C(5, 2) = 10, since 120 divided by 2 times 6 gives 10. Lottery scale: picking 6 numbers from 49 yields C(49, 6) = 13,983,816 possible tickets, one in nearly fourteen million for a straight pick.
Try a committee example with the actual inputs. A team of 30 volunteers holds 12 open spots, so type 30 into Choose from, 12 into How many?, and leave Without repetition on. The answer is C(30, 12) = 86,493,225, a figure nobody would attempt by listing groups. The formula is symmetric here: 30 choose 12 equals 30 choose 18, so a solver can attack whichever side has the shorter factorial run. Choosing a 12 person crew is also far lighter than arranging those 12 people into ordered slots, which would multiply the count by 12! and reach 479,001,600 times as many outcomes.
How to use the calculator
Enter the pool size in Choose from, for instance 49, then the pick size in How many?, such as 6, and keep the setting on Without repetition. Press Calculate to get the count. If order suddenly matters, like ranking podium finishes, combinations would undercount and permutations fit better, so confirm the question wording first.
Frequently Asked Questions
When does order matter between items?
Not for combinations: ABC and CBA count as one outcome. If arrangement carries meaning, such as first and second place, use permutations instead because they count ordered sequences separately.
What if repetition is allowed?
Allowing repeats replaces the formula with C(n + k - 1, k). Two scoops from 5 flavors with repetition allowed gives 15 instead of 10, and the gap widens quickly.
How big is 52 choose 5?
C(52, 5) = 2,598,960 distinct poker hands. Divide favorable hands by that figure for exact game probabilities, which anchors most poker strategy derivations from the ground up.
Why does factorial division remove duplicates?
Each set of k items can be ordered k! ways, all identical as a combination. Dividing by k! removes that overcount, so every distinct selection is counted exactly once.
Can n be smaller than k?
No, so the calculator returns zero or an error. You cannot pick 7 items from a pool of 5, which is why a weird zero usually signals a swapped input.