Permutation

Before you rely on it

What to check

Ask whether rearranging the same chosen items counts as a different outcome.

The common error

Where people go wrong with permutation

Using the permutation formula for a committee or a hand of cards. Order is irrelevant in both, and the answer comes out inflated by a factor that grows with the selection size.

Worked through

Permutation in plain numbers

Eight runners compete and the top three finish in gold, silver and bronze order; the number of possible podiums is 8 × 7 × 6, or 336. Order matters, so each different arrangement counts. Now a four-digit PIN. If digits can repeat, there are 10 × 10 × 10 × 10 = 10,000 codes. If no digit may repeat, the count is 10 × 9 × 8 × 7 = 5,040, a permutation, and the ban on repeats cuts the guesses roughly in half; the formula is n factorial divided by (n − r) factorial. Each step leaves fewer choices, and the running product is the count.

Easily confused

Permutation and its neighbours

A permutation counts arrangements where order matters. A combination counts selections where it does not. Choosing 3 people out of 8 for a committee is a combination, 56 ways. Assigning them to president, treasurer and secretary is a permutation, 336 ways, and the permutation count is always larger by a factor of r factorial. Permutations with repetition, like a PIN, allow reuse and use powers instead of a falling product. Permutations of items with duplicates, such as letters in a word, divide by the repeats.

In real decisions

Where permutation turns up

Permutations count passwords, lock combinations, race results, seating plans, schedules and rankings. They underlie probability calculations for events where order matters. Programmers use them to generate test cases and to reason about search space. Security analysts count them to estimate how long guessing would take; a calculator with an nPr function computes the count directly.

Know the edges

Where permutation stops being reliable

The numbers explode quickly, so brute-force listing of permutations is impractical beyond small n; the formula assumes distinct items and no repeats unless adjusted. Real problems often have constraints that require care. A count of arrangements does not tell you probabilities unless the outcomes are equally likely, and check whether order and repetition apply before choosing a formula.

Quick answers

What people ask about permutation

What is the formula for permutations? nPr equals n factorial divided by (n − r) factorial. For choosing and ordering 3 from 8, it is 8 × 7 × 6, or 336.

How do repeated items change the count? If some items are identical, divide by the factorial of each repeat. The word LEVEL has 5 letters with two Ls and two Es, so 5 factorial divided by (2 factorial times 2 factorial) gives 30 arrangements.

When are repeats allowed? In a PIN or a code where symbols may repeat, the count is n raised to the power r. Four digits from ten give 10,000.

How do permutations relate to probability? The chance of a specific ordered outcome is one divided by the number of permutations when all are equally likely. Getting a specific podium among 336 is 1 in 336.