Multinomialteoremet

The multinomialteoremet, known in English as the multinomial theorem, is a generalization of the binomial theorem to sums of more than two variables. If x1, x2, ..., xm are variables and n is a nonnegative integer, then the expansion of (x1 + x2 + ... + xm)^n is given by a sum over all m-tuples of nonnegative integers (k1, k2, ..., km) whose sum is n:

(x1 + x2 + ... + xm)^n = sum_{k1+...+km=n} [n!/(k1! k2! ... km!)] x1^{k1} x2^{k2} ... xm^{km}.

The coefficients n!/(k1! k2! ... km!) are called multinomial coefficients. They count the number of ways to distribute n identical items into m labeled boxes with k1, k2, ..., km items in each box.

For m = 2, the formula reduces to the familiar binomial theorem. A concrete example is (x + y + z)^3, which expands to x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3y^2x + 3y^2z + 3z^2x + 3z^2y + 6xyz.

Applications of the multinomial theorem appear in combinatorics, algebra, and probability. In probability, the multinomial distribution describes the counts of outcomes from n independent trials with m possible categories, with probability vector p1, p2, ..., pm; the probability of a particular count vector (k1, ..., km) is n!/(k1!...km!) ∏ p_i^{k_i}. The theorem also underpins generating functions and various polynomial identities.