Permutations
A permutation is an ordered arrangement of objects selected from a set. The key word here is 'ordered': if you change the order of the objects, you get a different permutation. For example, the letters A, B, and C can be arranged as ABC, ACB, BAC, BCA, CAB, or CBA — each arrangement is a distinct permutation, giving us 6 total. Permutations appear whenever the sequence or position of items matters, such as assigning medals (gold, silver, bronze), creating passwords, or scheduling tasks. Permutations are a foundational tool in combinatorics, the branch of mathematics concerned with counting, arrangement, and combination. Understanding permutations allows you to answer questions like: 'In how many ways can 10 runners finish a race in first, second, and third place?' or 'How many 4-digit PIN codes can be formed from the digits 0–9 if no digit repeats?' These questions arise naturally in probability, computer science, cryptography, and everyday decision-making. There are two main types of permutations you will encounter: permutations of all n objects in a set, and permutations of r objects chosen from a set of n (called n-permute-r, or P(n, r)). Both rely on the Fundamental Counting Principle and the concept of the factorial function. Mastering permutations will also lay the groundwork for understanding combinations, where order does not matter.
- Understand the Fundamental Counting Principle
- Learn the Factorial Function
- Apply the Permutation Formula P(n, r)
- Handle Permutations with Repetition
- Handle Permutations of Non-Distinct Objects
Understand the Fundamental Counting Principle
Before using the permutation formula, understand why it works. If you have n objects to fill k positions, the first position can be filled in n ways, the second in (n−1) ways (since one object is used), the third in (n−2) ways, and so on. You multiply all these choices together. This is the Multiplication (Fundamental Counting) Principle: the total number of outcomes equals the product of the choices at each stage.
Learn the Factorial Function
The factorial of a positive integer n, written n!, is the product of all positive integers from 1 to n: n! = n × (n−1) × (n−2) × ⋯ × 2 × 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1. Factorials grow extremely quickly: 10! = 3,628,800. The factorial is the engine inside every permutation calculation.
Apply the Permutation Formula P(n, r)
When you want to arrange r objects chosen from n distinct objects (where order matters and repetition is not allowed), use the formula P(n, r) = n! / (n − r)!. This formula works because you start with n choices, then n−1, down to (n−r+1), and the division by (n−r)! cancels out the factors you don't use. When r = n (arranging all objects), P(n, n) = n! / 0! = n!, which matches the full arrangement count.
Handle Permutations with Repetition
Sometimes objects may be repeated. If you are forming r-length arrangements from n objects where repetition IS allowed (e.g., PIN codes), the count is simply n^r — each of the r positions has n independent choices. For example, a 4-digit PIN from digits 0–9 with repetition gives 10^4 = 10,000 possibilities. This is distinct from the standard permutation formula, so always check whether repetition is permitted.
Handle Permutations of Non-Distinct Objects
If some objects are identical, arrangements that look different by position but contain the same objects in the same order must not be double-counted. The formula is n! / (n₁! × n₂! × ⋯ × nₖ!), where n₁, n₂, …, nₖ are the counts of each repeated object. For example, the word MISSISSIPPI has 11 letters (1 M, 4 I, 4 S, 2 P), giving 11! / (1! × 4! × 4! × 2!) = 34,650 distinct arrangements.
The Key Equation
- P(n, r) — Number of permutations (dimensionless (count)): The total number of distinct ordered arrangements of r items selected from n distinct items.
- n — Total number of objects (dimensionless (count)): The size of the full set from which objects are being selected.
- r — Number of objects selected (dimensionless (count)): The number of positions to fill; must satisfy 0 ≤ r ≤ n.
- n! — n factorial (dimensionless): The product n × (n−1) × (n−2) × ⋯ × 1, representing all ways to arrange n objects.
- (n - r)! — (n minus r) factorial (dimensionless): Cancels the factors not used when only r of the n objects are arranged.
Worked Example
Problem. A school committee must select a President, Vice-President, and Secretary from a group of 8 students. No student can hold more than one office. In how many ways can the three positions be filled?
Answer. 336 ordered arrangements
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
Leibniz made significant early contributions to combinatorics, including systematic work on permutations and combinations in his 1666 dissertation 'Dissertatio de Arte Combinatoria.' He explored how arrangements of symbols could generate new ideas and logical structures, laying conceptual groundwork for modern combinatorial mathematics. He introduced notations and counting arguments that influenced how mathematicians formalised permutation theory. Leibniz co-invented calculus independently of Isaac Newton, leading to one of history's most famous scientific disputes — but he also found time to design mechanical calculators and dream of a universal logical language for all human thought.
Related Concepts
- Combinations — Combinations count selections where order does NOT matter, contrasting directly with permutations where order is essential. Learning both clarifies when to apply each formula.
- The Factorial Function — Factorials are the mathematical engine of the permutation formula; understanding n! is prerequisite to deriving and applying P(n, r) correctly.
- Fundamental Counting Principle — The Multiplication Principle explains why you multiply choices at each step, forming the conceptual foundation from which the permutation formula is derived.
- Basic Probability — Permutations are used to count equally likely outcomes in probability calculations, for example, computing the probability that a randomly ordered list is in a specific sequence.