Multinomial coefficient

Definition

A multinomial coefficient is a generalization of the binomial coefficient that counts the number of ways to divide a set of n objects into k distinct groups, where each group contains a specified number of objects. It is represented mathematically as $$\frac{n!}{n_1!n_2!...n_k!}$$, where n is the total number of objects and each n_i represents the size of group i. This concept is crucial in combinatorics, especially when dealing with permutations and combinations of objects that can be categorized into multiple classes.

5 Must Know Facts For Your Next Test

1. The multinomial coefficient counts the ways to distribute n identical items into k distinct categories, where each category can have a specified number of items.
2. The formula for the multinomial coefficient is derived from the factorial function and includes dividing by the factorials of the sizes of each group.
3. Multinomial coefficients play a critical role in probability theory and statistical distributions, especially in scenarios involving multiple categories.
4. The sum of multinomial coefficients for all possible distributions equals 1, representing the total probability in scenarios involving multiple outcomes.
5. Multinomial coefficients can be used to expand expressions like $$(x_1 + x_2 + ... + x_k)^n$$ using the multinomial theorem.

Review Questions

• How do multinomial coefficients extend the concept of binomial coefficients in combinatorial counting?
  • Multinomial coefficients generalize binomial coefficients by allowing for the division of n objects into more than two groups. While a binomial coefficient counts the ways to choose k objects from n without regard to order, multinomial coefficients consider multiple categories and their specific sizes. The formula $$\frac{n!}{n_1!n_2!...n_k!}$$ accounts for all group sizes, making it useful in more complex distribution problems.
• In what situations would you prefer using multinomial coefficients over simple permutations or combinations?
  • Multinomial coefficients are preferred when dealing with problems that involve distributing objects into multiple distinct categories with fixed quantities. For instance, if you need to allocate 10 identical candies into 3 different bags with 4, 3, and 3 candies respectively, using multinomial coefficients gives a precise count of arrangements that meet those criteria. In contrast, simple permutations or combinations would not suffice since they typically handle fewer categories.
• Evaluate how the multinomial theorem uses multinomial coefficients in expanding polynomial expressions and its significance in combinatorics.
  • The multinomial theorem states that $$(x_1 + x_2 + ... + x_k)^n$$ can be expanded using multinomial coefficients, which represent the number of ways to choose terms from each category during expansion. Each term in the expansion corresponds to a specific distribution of powers across the variables, and thus multinomial coefficients provide both combinatorial significance and practical computational utility. This application links combinatorics with algebra, enabling efficient calculations in various mathematical fields.

Related terms

binomial coefficient: A binomial coefficient represents the number of ways to choose k elements from a set of n elements and is denoted as $$\binom{n}{k}$$.

permutations: Permutations refer to the different arrangements of a set of objects, where the order matters. The number of permutations of n objects taken r at a time is given by $$P(n, r) = \frac{n!}{(n - r)!}$$.

combinations: Combinations count the ways to select items from a larger pool without regard to the order of selection, denoted as $$\binom{n}{r}$$.