Introduction to Permutations
Learn how to count arrangements where order matters using permutations.
Definition
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Worked Examples
In how many ways can 4 students (Anna, Ben, Carla, David) line up for a photo?
Identify what we're counting
We need to arrange all 4 students in a line where position matters → This is a permutation of 4 objects
Count choices for each position
1st position: 4 choices\\2nd position: 3 remaining\\3rd position: 2 remaining\\4th position: 1 remaining →
Calculate using factorial
→
Interpret the result
There are 24 different ways to arrange 4 students → 24 arrangements
Answer: There are different ways to line up for the photo.
Common Mistakes
Confusing permutations and combinations
Why it's wrong: Permutations count arrangements where ORDER MATTERS. Combinations count selections where order doesn't matter.
Correct: Ask yourself: "Does rearranging change the outcome?" If selecting a committee (no positions), order doesn't matter = combination. If electing officers, order matters = permutation.
Using the wrong formula direction: instead of
Why it's wrong: The larger factorial goes on top because we start with more choices.
Correct: — remember: is always larger, so it goes in the numerator.
Forgetting that
Why it's wrong: When arranging all n objects, we use
Correct: By definition, . This makes the formula work when .
Interactive Sandbox
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Practice Problems
18 problemsWhat is (3 factorial)?
Why It Matters
- Passwords: How many 4-digit PINs use different digits?
- Races: How many ways can gold, silver, bronze medals be awarded?
- Scheduling: In how many orders can 5 tasks be completed?
- Seating: How many ways can students sit in a row?
Real World Applications
Password Security
Understanding permutations helps us calculate password strength and security.
Example:
A 4-digit PIN using digits 0-9 without repetition has possibilities.
A website requires a 3-character password using only uppercase letters (A-Z), with no letter repeated.
How many possible passwords are there?
Step 1: Write the mathematical expression
Calculate :
Sports Tournament Brackets
Permutations determine possible outcomes in competitions where finishing position matters.
Example:
In a 6-team playoff, the possible final standings are different orders.
An Olympic swimming final has 8 swimmers. Medals are awarded for 1st, 2nd, and 3rd place.
How many different medal outcomes are possible?
Step 1: Write the mathematical expression
Use :
Key Takeaways
- 1A permutation is an arrangement where order matters
- 2To arrange all objects: use
- 3To arrange objects from objects: use
- 4Quick method: multiply for terms
- 5Remember: by definition
Frequently Asked Questions
Glossary
- Permutation
- An arrangement of objects where order matters
- Factorial
- The product of all positive integers up to n, written as
- P(n,r)
- The number of permutations of r objects selected from n objects
- Distinct objects
- Objects that are all different from each other
Formula Card
Factorial
Product of all positive integers up to n
Permutation
Arrangements of r objects from n
Special cases
Important values to remember