Permutations Summary

Combinatorics · Axiom Academy

Let's review the fundamental concepts of ordering and arranging objects. Definition: A permutation is an ordered arrangement of objects where the sequence matters Key Insight: ABC is different from BAC - order creates distinct outcomes Example: Three people in a line can be arranged in 6 different ways Why It Matters: Permutations model situations where position and sequence are critical Notation: n! represents "n factorial" Meaning: The product of all positive integers from 1 to n Formula: n! = n × (n-1) × (n-2) × ... × 2 × 1 Special Case: By convention, 0! = 1 Importance: Counts all possible orderings of n distinct objects Full Permutations: All n objects arranged - computed as n! Partial Permutations: Select and arrange r from n objects - use P(n,r) Circular Permutations: Arrangements in a circle - computed as (n-1)! due to rotational symmetry Permutations with Repetition: When objects can be reused - computed as n r Multiset Permutations: When some objects are identical - divide by factorials of repetition counts: n!/(n₁!×n₂!×...×nₖ!) Derangements: Permutations where no object appears in its original position - denoted !n Example Recap: Selecting and Arranging Committee Officers Problem Setup: From 10 people, select a president, vice president, and secretary (3 different people, order matters) Identify the Type: This is a partial permutation because we're choosing and arranging r = 3 people from n = 10 total Apply the Formula: P(10,3) = 10!/(10-3)! = 10!/7!

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