Question: A catering service offers 9 appetizers, 11 main courses, and 5 desserts. A banquet committee is to…

A catering service offers 9 appetizers, 11 main courses, and 5 desserts. A banquet committee is to select 8 appetizers, 10 main courses, and 4 desserts. How many ways can this be done? … There are possible ways this can be done.

Solution

The problem involves choosing a certain number of items from a larger set. We will use combinations to find the total number of ways to select the appetizers, main courses, and desserts separately, and then multiply these results together. First, calculate the number of ways to choose 8 appetizers from 9 options. Use the combination formula \( \binom{n}{k} \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose. \[ \binom{9}{8} = \frac{9!}{8!(9-8)!} = \frac{9!}{8! \cdot 1!} = 9 \] Next, calculate the number of ways to choose 10 main courses from 11 options. \[ \binom{11}{10} = \frac{11!}{10!(11-10)!} = \frac{11!}{10! \cdot 1!} = 11 \] Then, calculate the number of ways to choose 4 desserts from 5 options. \[ \binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5!}{4! \cdot 1!} = 5 \] Now multiply the results from each category together to find the total number of ways to make the selections. \[ 9 \times 11 \times 5 = 495 \] There are 495 possible ways this can be done.