Multinomial coefficient (n; k₁,...,kₘ) = n!/(k₁!·...·kₘ!). Counts ways to divide n objects into groups of size k₁,...,kₘ. Binomial is the m=2 case. Multinomial theorem: (x₁+...+xₘ)^n = Σ(n;k₁,...,kₘ)·x₁^k₁·...·xₘ^kₘ.
How to Use
Enter n and group sizes:
- Coefficient: The multinomial value
- Factorials: Breakdown
- Verify: Sum of kᵢ = n
Multinomial Theorem
(x₁+x₂+...+xₘ)^n = Σ (n;k₁,...,kₘ)·∏xᵢ^kᵢ summed over all non-negative kᵢ with Σkᵢ=n. The binomial theorem (a+b)^n is the special case m=2.
Applications
- Counting anagrams: MISSISSIPPI → 11!/(1!4!4!2!) = 34650
- Multinomial distribution in statistics
- Combinatorial identities
Step-by-Step Instructions
- 1Enter total n.
- 2Enter group sizes.
- 3Verify sum.
- 4Compute coefficient.
- 5View breakdown.