Derangements
Permutations with no fixed points.
Key idea
Permutations with no fixed points.
In depth
A derangement is a permutation with no fixed points. Dₙ = n! Σ (−1)ᵏ/k!, approximately n!/e. The chance a random permutation is a derangement tends to 1/e ≈ 0.368.
Worked example
D₄ = 9: the number of ways 4 hats can be returned so that nobody gets their own.
Formula
Real-world uses
- Counting ways letters can go to the wrong envelopes
- Gift-exchange problems where nobody gets their own gift
Common mistake
Subtracting only the single-fixed-point cases and ignoring overlaps.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Derangements?
Permutations with no fixed points. It is a topic at the expert level in Discrete Math & Number Theory.
What is the formula for Derangements?
The key formula is Dₙ = n!Σ(−1)ᵏ/k!.
Where is Derangements used in real life?
Real-world examples: Counting ways letters can go to the wrong envelopes; Gift-exchange problems where nobody gets their own gift.
What should I learn before Derangements?
Make sure you are comfortable with Multinomial Coefficients, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Derangements?
Subtracting only the single-fixed-point cases and ignoring overlaps.
How can I practise Derangements?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Derangements?
Move on to Catalan Numbers next.
Related expert topics in Discrete Math & Number Theory
- Strong Induction – Assume all earlier cases to prove the next.
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Cardinality of Sets – Comparing sizes of infinite sets.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Cantor’s Diagonal Argument – Proof that real numbers are uncountable.