Permutation

Permutation: Definition, Formula, and Examples

A permutation is a mathematical calculation that determines the total number of unique ways a specific set of objects can be arranged, where the order of arrangement matters.

1. Definition of Permutation

A permutation is an ordered arrangement of all or part of a set of items. Unlike combinations, changing the sequence of the items creates a completely new permutation. For example, the arrangement $(A,B,C)$ is counted as a distinct permutation from $(C,B,A)$.

2. The Formula

When choosing and arranging $r$ unique items from a total pool of $n$ items, the formula is:

$${}^n{P}_r=\frac{n!}{(n-r)!}$$

Where:

• $n$ is the total number of items in the set.
• $r$ is the number of items chosen for the arrangement.
• $!$ denotes a factorial (e.g., $4!=4\times 3\times 2\times 1=24$).

3. Step-by-Step Examples

Example 1: Arranging a Full Set ($n=r$)

Suppose you want to arrange 3 books (A, B, and C) on a shelf.

• Formula application: Because you are arranging all 3 items, $n=3$ and $r=3$. This simplifies to $n!$ or $3!$.
• Calculation: $3!=3\times 2\times 1=6$
• The 6 possible permutations: ABC, ACB, BAC, BCA, CAB, CBA.

Example 2: Arranging a Subset ($n>r$)

Suppose a race has 5 runners, and you want to find out how many different ways the Gold, Silver, and Bronze medals (top 3 spots) can be awarded.

• Identify values: Total runners $n=5$. Medaled positions $r=3$.
• Apply formula: ${}^5{P}_3=\frac{5!}{(5-3)!}=\frac{5!}{2!}$
• Calculation: $\frac{5\times 4\times 3\times 2\times 1}{2\times 1}=5\times 4\times 3=60$
• Result: There are 60 different ways to award the podium medals.

4. Comparison Summary

Concept	Order Matters?	Example
Permutation	Yes	Creating a lock passcode ($1\text{-}2\text{-}3$ is different from $3\text{-}2\text{-}1$)
Combination	No	Selecting a team of 3 people (Alice and Bob is the same team)

5. Core Concept Summary

A permutation calculates the number of ways to arrange items where order is crucial. If you change the sequence, you create a brand new outcome.