Permutations

Definition

Permutations are methods for calculating the number of ways to select and arrange a subset of items from a larger set where the order of selection matters.

Why It Matters

Permutations are critical when dealing with ordered systems—like passwords, schedules, or genetic sequences. Confusing permutations with combinations will cause you to massively underestimate the complexity and total number of possibilities in any order-dependent system.

Core Concepts

• Without Repetition: $P(n, r) = \frac{n!}{(n-r)!}$

  • How to read: “The number of permutations of n items taken r at a time is equal to n factorial divided by the quantity n minus r factorial.”
  • Meaning: Ordered arrangements of $r$ items from $n$—order matters, no repeats.

• With Repetition: $n^r$

  • How to read: “The value is n raised to the power of r.”
  • Meaning: $r$ independent choices from $n$ options each—order matters, repeats allowed.

• Nondistinct Objects: $\frac{n!}{n_1! n_2! \dots n_k!}$

  • How to read: “The total permutations is n factorial divided by the product of the factorials of the counts of each repeating item.”
  • Meaning: Permutations of $n$ objects with identical groups—divide out overcounting (used for words with repeating letters).

Connected Concepts

• Combinations
• Multiplication Principle
• Addition Principle
• Inclusion-Exclusion Principle
• Probability Models: Equally Likely Outcomes
• Map-Territory Relation