Calculate permutations, combinations, factorials, and multinomial coefficients with visual counting diagrams and Pascal triangle.
Order is the decision that changes most answers in this calculator. Enter a total count and, where needed, a selection size or group sizes to evaluate factorials, permutations, combinations, and multinomial coefficients. Visual counting diagrams help distinguish arrangements from selections, while Pascal triangle shows how combination values relate. The most common mistake is using a permutation when rearranging the chosen items should not create a new outcome.
A concrete input and expected output from the current implementation.
Input
Combination: n = 5, r = 2
Expected output
10
Choosing 2 items from 5 without regard to order gives C(5,2) = 5! / (2! * 3!) = 10. Each pair is counted once, so AB and BA are the same selection.
Factorial counts orderings of all n distinct items: n!. A permutation counts ordered selections using n! / (n-r)!, while a combination counts unordered selections using n! / (r!(n-r)!). These formulas use nonnegative integer counts, and 0! equals 1. Changing whether order matters can therefore change the result substantially.
The counting diagrams provide a visual interpretation of possible choices or arrangements. Pascal triangle presents binomial coefficients, with the entries in row n corresponding to C(n,0) through C(n,n) when the top is labeled row 0. Adjacent entries also satisfy C(n,r) = C(n-1,r-1) + C(n-1,r).
Use a permutation when changing the order creates a different result, such as first, second, and third place. Use a combination when only membership matters, such as choosing three committee members. For n = 5 and r = 2, permutations give 20 ordered pairs, while combinations give 10 unordered pairs.
Yes, when the multiplicity of each object type is specified. For group sizes k1 through km totaling n, the count is n! divided by k1! through km!. This counts distinct arrangements of a fixed multiset. It does not by itself model unrestricted repeated selection, where an item may be chosen again any number of times.
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