Permutation & Combination Calculator

Understanding Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics, the branch of mathematics that deals with counting. They answer the question: in how many ways can you select or arrange items from a larger set? The key distinction is whether order matters.

A permutation counts the number of ways to arrange r items chosen from n items where the order of arrangement matters. Think of it like assigning first, second, and third place in a race: who finishes where matters. The formula is nPr = n! / (n-r)!, where n! (n factorial) means n multiplied by every positive integer below it. For example, arranging 3 books from a shelf of 10 gives 10P3 = 720 distinct arrangements.

A combination counts selections where order does not matter. Choosing 3 members for a committee from 10 people is a combination problem: the same 3 people form the same committee regardless of selection order. The formula is nCr = n! / (r! x (n-r)!), which divides the permutation count by r! to eliminate duplicate orderings. Combinations appear in lottery probability, poker hands, and statistical sampling. This calculator computes both values instantly and displays the step-by-step formula breakdown so you can verify and learn the process.