Permutation Definition

Combinatorics · Axiom Academy

Understanding ordered arrangements and why sequence matters in combinatorics A permutation is an ordered arrangement of objects selected from a set. The key characteristic is that order matters —changing the sequence creates a different permutation. Consider three letters: A, B, C. The arrangement ABC is different from BAC or CAB because the order is different. Each distinct ordering is a separate permutation. We use special notation to represent permutations of r objects selected from n total objects: P(n,r) - Standard mathematical notation nPr - Calculator and textbook notation This reads as "the number of permutations of n objects taken r at a time" or "r-permutations of n objects." 5 choices for the first position 4 choices for the second position 3 choices for the third position 3. Why Order Matters in Permutations The fundamental characteristic of permutations is that position is significant . The same objects in different positions create distinct permutations. Consider selecting 2 letters from A, B, C : This gives us 6 total permutations: AB, BA, AC, CA, BC, CB 4. Permutations: All Objects vs. Some Objects We distinguish between two important cases: 1. Permutation of all n objects: P(n,n) = n! This is simply n factorial—arranging all available objects in order. 2. Permutation of r objects from n: P(n,r) where r < n This is an r-permutation—selecting and arranging r objects from n available objects. All objects (r=4): P(4,4) = 4! = 24 ways to arrange all 4 letters

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