/mʌltɪˈnoʊmiəl kəˈɪfɪʃənt/
A multinomial coefficient is a generalization of binomial coefficients. It is a coefficient that appears in the multinomial expansion of a power of a sum. The multinomial coefficient represents the number of ways to partition ( n ) objects into ( k ) groups of specified sizes. Formally, the multinomial coefficient for ( n ) total objects divided into ( k ) groups of sizes ( n_1, n_2, \dots, n_k ) (where ( n_1 + n_2 + \ldots + n_k = n )) is given by:
[ \frac{n!}{n_1! n_2! \ldots n_k!} ]
In terms of usage, the term "multinomial coefficient" is common in combinatorial mathematics and is primarily used in written context, particularly in academic papers or textbooks related to probability and statistics.
Многомерный коэффициент помогает вычислить различные способы расположения нескольких типов объектов.
In statistics, the multinomial coefficient is used to find probabilities in categorical distributions.
В статистике многомерный коэффициент используется для нахождения вероятностей в категориальных распределениях.
You can compute the multinomial coefficient for a given set of group sizes using its formula.
The phrase "multinomial coefficient" is not widely used in idiomatic expressions since it is a specific mathematical term. However, in academic and professional settings, you may hear variations in discussions about combinatorial mathematics and statistics.
Для вычисления вероятностей нам часто нужно применять теорему многономов.
The arrangement of cards can be determined by the multinomial coefficients derived from the set's values.
Расположение карт можно определить с помощью многомерных коэффициентов, полученных из значений набора.
Combinatorial designs require a deep understanding of multinomial coefficients for their formulation.
The term "multinomial" is derived from the combination of the prefix "multi-", meaning many, and the word "nomial", which comes from the Latin "nomen" meaning name. The word "coefficient" comes from the Latin "coefficientem", which means to work together. Thus, the term essentially refers to the 'many names' (different groups or arrangements) that can be represented in combinations.