Multinomial theorem

From Academic Kids

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

<math>(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} <math>

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

<math> {n \choose k_1, k_2, k_3, \ldots, k_m}
= \frac{n!}{k_1! k_2! k_3! \cdots k_m!}<math>

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.

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