gen-define-the-multinomial-coefficient-and-give-one-short-worked-99b4

Define the multinomial coefficient and give one short worked example.

# Multinomial Coefficient ## Definition The **multinomial coefficient** is defined as: $$\binom{n}{k_1, k_2, \ldots, k_m} = \frac{n!}{k_1! \cdot k_2! \cdot \ldots \cdot k_m!}$$ where $n = k_1 + k_2 + \cdots + k_m$. It counts the number of ways to divide $n$ distinct objects into $m$ groups of sizes $k_1, k_2, \ldots, k_m$. ## Worked Example **Question:** In how many ways can we arrange the letters in the word "BANANA"? **Solution:** The word BANANA has: - 6 letters total - 3 A's - 2 N's - 1 B Using the multinomial coefficient: $$\binom{6}{3, 2, 1} = \frac{6!}{3! \cdot 2! \cdot 1!} = \frac{720}{6 \cdot 2 \cdot 1} = \frac{720}{12} = 60$$ There are **60** distinct arrangements of the letters in BANANA.