multinomial theorem

In mathematics, the multinomial formula is an expression of a power of a sum in terms of powers of the addends. For any positive integer m and any nonnegative integer n, the multinomial formula is

(x_1 + x_2 + x_3 + \cdots + x_m)^n

  = \sum_{k_1,k_2,k_3,\ldots,k_m} {n \choose k_1, k_2, k_3, \ldots, k_m}   x_1^{k_1} x_2^{k_2} x_3^{k_3} \cdots x_m^{k_m}

The summation is taken over all combinations of the indices k₁ through k_m such that k₁ + k₂ + k₃ + ... + k_m = n; some or all of the nonnegative indices may be zero. The numbers

{n \choose k_1, k_2, k_3, \ldots, k_m}

  = \frac{n!}{k_1! k_2! k_3! \cdots k_m!}

are the multinomial coefficients. The multinomial coefficients have a direct combinatorial interpretation, as the number of ways of depositing n distinguished objects in m bins, with k₁ in the first, and so on. This is an equivalent assertion. The binomial theorem and binomial coefficient are special cases, for m = 2, of the multinomial formula and multinomial coefficient, respectively. Therefore this is also called the multinomial theorem.

Multinomial Theorem

See also