Permutation & Combination Calculator

Permutation and Combination: The Basics of Counting

Permutations and combinations are fundamental concepts in combinatorics, a branch of mathematics concerning the study of counting, arrangement, and combination.

Permutation (Order Matters)

A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. For example, if you have three letters A, B, and C, the possible permutations of selecting two letters are AB, AC, BA, BC, CA, CB. Key Concept: Order matters! "AB" is different from "BA".

Formula

The number of permutations of n objects taken r at a time is given by:

A(n, r) = n! / (n-r)!

Combination (Order Doesn't Matter)

A combination is a selection of items from a collection, such that the order of selection does not matter. Using the same example of letters A, B, and C, the combinations of selecting two letters are AB, AC, BC. Key Concept: Order does not matter! "AB" is the same as "BA".

Formula

The number of combinations of n objects taken r at a time is given by:

C(n, r) = n! / (r! * (n-r)!)

Key Differences

Real World Examples

1. Passwords: A password "1234" is different from "4321". This is a permutation.
2. Lottery: If you pick numbers 1, 2, 3 and the winning numbers are 3, 2, 1, you still win. This is a combination.