Problem 1 - Thinking Mathematically

Question

A restaurant offers eight appetizers and ten main courses. In how many ways can a person order a two-course meal?

Step-by-Step Solution

Answer

80

Step 1 Count of Choices for Appetizers

There are 8 different appetizers to choose from. So there are 8 ways to choose an appetizer.

Step 2 Count of Choices for Main Courses

There are 10 different main courses to choose from. So there are 10 ways to choose a main course.

Step 3 Calculate the total number of choices

This is a combination (without repetition) situation, so the total number of ways to choose two courses (an appetizer and a main course) is the product of the number of choices for each course. Multiply the number of ways to choose an appetizer (8) by the number of ways to choose a main course (10). \(8 \times 10 = 80 \)

Key Concepts

Permutations

Permutations refer to the arrangement of items in a specific order. When dealing with permutations, the order is crucial, as each arrangement counts as a different permutation. Consider a simpler example: arranging three distinct letters, such as A, B, and C.

• The possible arrangements are: ABC, ACB, BAC, BCA, CAB, CBA, which total to six different permutations.
• The formula used for permutations of n distinct items is n! (n factorial), which means multiplying the sequence of natural numbers up to n.

In the context of meals, permutations might be applied if the order in which courses are served mattered, which is not the case in our exercise. Here the sequence of courses is fixed, meaning permutations aren’t applicable directly, but it's important to understand the concept to distinguish it from combinations.

Combinations

Combinations focus on the selection of items regardless of order, which makes them different from permutations. The order here does not affect the combination, meaning that picking items in different sequences counts as one.Imagine you are choosing two flavors for an ice cream cone where the order of the flavors doesn't matter.

• If you have chocolate, vanilla, and strawberry to choose from, chocolate followed by vanilla is the same as vanilla followed by chocolate.
• For such scenarios, you calculate combinations using the formula \( \binom{n}{r} = \frac{n!}{r! (n-r)!} \) where \(n\) is the total number of items, and \(r\) is the number of items you are choosing.

In the restaurant problem, although you are making two selections (an appetizer and a main course), it's not a combination because each choice only happens once, with no repetition in the group of items. Choices are independent and hence, a simple multiplication suffices to find the total number of combinations.

Mathematical Reasoning

Mathematical reasoning is the logical thought process used to solve problems, which involves identifying the type of problem and selecting the right approach or formula.In the provided exercise, strong reasoning involves recognizing that selecting an appetizer and a main course are independent events. Since each choice is independent:

• The total number of possible two-course meals can be calculated by multiplying the number of appetizers by the number of main courses.
• This multiplication approach is derived from the fundamental counting principle, which states that if an event can happen in \(m\) ways and another independent event can occur in \(n\) ways, then there are \(m \times n\) ways in which both events can happen.

The exercise's solution is not about permutations or combinations in a traditional sense, but about understanding that multiplication here allows us to determine all possible meal combinations effectively.