Multinomial Theorem: Expanding Expressions with Multiple Terms

Understand the multinomial theorem, a generalization of the binomial theorem for expanding expressions with more than two terms. This guide explains the multinomial theorem, its formula, multinomial coefficients, and provides examples illustrating its application.

Multinomial Theorem in Discrete Mathematics

What is the Multinomial Theorem?

The multinomial theorem is a generalization of the binomial theorem. The binomial theorem helps expand expressions like (x + y)ⁿ. The multinomial theorem extends this to expressions with more than two terms, such as (x₁ + x₂ + ... + x_k)ⁿ.

The Multinomial Theorem Formula

The multinomial theorem states that for any positive integer n and any non-negative integers n₁, n₂, ..., n_k such that n₁ + n₂ + ... + n_k = n:

(x₁ + x₂ + ... + x_k)ⁿ = Σ (n! / (n₁!n₂!...n_k!))x₁^n₁x₂^n₂...x_k^n_k

The summation is taken over all possible combinations of non-negative integers n₁, n₂, ..., n_k that sum to n. The terms (n! / (n₁!n₂!...n_k!)) are called multinomial coefficients.

Multinomial Coefficients

Multinomial coefficients are generalizations of binomial coefficients. They represent the number of ways to arrange n objects into k distinct groups, where group i contains n_i objects. The formula for a multinomial coefficient is:

n! / (n₁!n₂!...n_k!)

Examples: Applying the Multinomial Theorem

Example 1: Expanding a Trinomial Cubed

(The expansion of (x + y + z)³ is shown in the original text and should be included here.)

Example 2: Expanding a Trinomial to the Fourth Power

Let's expand (x + y + z)⁴. A brute-force approach (multiplying the expression out repeatedly) would result in many terms that simplify to the same expression. The multinomial theorem helps us find the simplified, concise expansion more efficiently.

(The expansion of (x + y + z)⁴ is shown in the original text and should be included here.)

Deriving the Multinomial Theorem

We can derive the multinomial theorem using nested summations. The process involves:

1. Starting with a nested summation that accounts for all possible combinations of powers of the terms.
2. Adding a filter (Kronecker delta function) to select only the terms where the sum of the exponents is equal to n.
3. Deriving the correct multinomial coefficient for each term.

(The step-by-step derivation, involving the Kronecker delta function and the calculation of multinomial coefficients, is provided in the original text and should be included here.)

Generalization to n Terms

The multinomial theorem can be generalized to an expression with any number of terms. For example, for four terms, the expansion would be:

(x₁ + x₂ + x₃ + x₄)ⁿ = Σ [n!/(n₁!n₂!n₃!n₄!)]x₁^n₁x₂^n₂x₃^n₃x₄^n₄

Conclusion

The multinomial theorem provides a powerful and efficient way to expand expressions raised to a power. It's a valuable tool in algebra and has applications in probability and statistics (multinomial distribution).