Multinomial Coefficients
Ways to split objects into labelled groups.
Key idea
Ways to split objects into labelled groups.
In depth
The number of ways to split n objects into groups of sizes k₁, ..., kᵣ is n!/(k₁!...kᵣ!). It is the coefficient in the multinomial expansion of (x₁ + ... + xᵣ)ⁿ.
Worked example
Dividing 10 students into groups of 5, 3, 2: 10!/(5!3!2!) = 2520.
Formula
Real-world uses
- Dividing a group of 10 workers into labelled teams of different sizes
Common mistake
Using it when groups of equal sizes are unlabelled; divide by symmetric arrangements.
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 Multinomial Coefficients?
Ways to split objects into labelled groups. It is a topic at the expert level in Discrete Math & Number Theory.
What is the formula for Multinomial Coefficients?
The key formula is n!/(k₁!k₂!…kₘ!).
Where is Multinomial Coefficients used in real life?
Real-world examples: Dividing a group of 10 workers into labelled teams of different sizes.
What should I learn before Multinomial Coefficients?
Make sure you are comfortable with Binomial Theorem (Discrete), the previous topic in Discrete Math & Number Theory.
What is a common mistake with Multinomial Coefficients?
Using it when groups of equal sizes are unlabelled; divide by symmetric arrangements.
How can I practise Multinomial Coefficients?
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 Multinomial Coefficients?
Move on to Derangements 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.