Multinomial Theorem

Combinatorics · Axiom Academy

Extending the Binomial Theorem to Multiple Terms When we expand (x₁ + x₂ + ... + xₖ)ⁿ, we get a sum of terms where each term is a product of powers of the variables, and the exponents must add up to n. The sum is taken over all sequences of non-negative integers (n₁, n₂, ..., nₖ) such that n₁ + n₂ + ... + nₖ = n. The multinomial coefficient counts the number of ways to arrange n objects where we have n₁ of type 1, n₂ of type 2, and so on. This is a direct generalization of the binomial coefficient. Notation: We write this as (n; n₁, n₂, ..., nₖ) or sometimes as a multinomial symbol. 3. Combinatorial Interpretation The multinomial coefficient has a beautiful combinatorial meaning: it counts the number of ways to distribute n distinct items into k groups with group sizes n₁, n₂, ..., nₖ. 4. Example: Expanding (x + y + z)³ Let's apply the multinomial theorem to expand (x + y + z)³. We need to find all ways that three exponents can sum to 3. 5. Connection to Binomial Theorem The multinomial theorem truly generalizes the binomial theorem. When k = 2, we recover exactly the binomial expansion. Since n₁ + n₂ = n, we have n₂ = n - n₁, and the multinomial coefficient simplifies to the binomial coefficient C(n, n₁).

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