Multinomial Theorem

multinomial theorem, in algebra, a generalization of the binomial theorem to more than two variables. In statistics, the corresponding multinomial series appears in the multinomial distribution, which is a generalization of the binomial distribution.

The multinomial theorem provides a formula for expanding an expression such as (x₁ + x₂ +⋯+ x_k)ⁿ for integer values of n. In particular, the expansion is given by where n₁ + n₂ +⋯+ n_k = n and n! is the factorial notation for 1 × 2 × 3 ×⋯× n.

For example, the expansion of (x₁ + x₂ + x₃)³ is x₁³ + 3x₁²x₂ + 3x₁²x₃ + 3x₁x₂² + 3x₁x₃² + 6x₁x₂x₃ + x₂³ + 3x₂²x₃ + 3x₂x₃² + x₃³.