Problem 9 - Thinking Mathematically

Question

A restaurant offers the following limited lunch menu.$$ \begin{array}{|l|l|l|l|} \hline \text { Main Course } & \text { Vegetables } & \text { Beverages } & \text { Desserts } \\ \hline \text { Ham } & \text { Potatoes } & \text { Coffee } & \text { Cake } \\\ \hline \text { Chicken } & \text { Peas } & \text { Tea } & \text { Pie } \\ \hline \text { Fish } & \text { Green beans } & \text { Milk } & \text { Ice cream } \\ \hline \text { Beef } & & \text { Soda } & \\ \hline \end{array} $$ If one item is selected from each of the four groups, in how many ways can a meal be ordered? Describe two such orders.

Step-by-Step Solution

Answer

The total number of ways to order a meal is 144. Two such orders could be: 1)Ham, Potatoes, Coffee, and Cake 2)Fish, Green Beans, Milk, and Ice Cream.

Step 1: Counting Principle

The counting principle states the total possible ways of selecting items is the product of the number of options in each category. In this case, for the main course, there are 4 choices (Ham, Chicken, Fish, Beef), for the vegetables there are 3 choices (Potatoes, Peas, Green Beans), for the beverages there are 4 choices (Coffee, Tea, Milk, Soda), and for the desserts there are 3 choices (Cake, Pie, Ice Cream). The total number of ways a meal can be ordered is \(4 * 3 * 4 * 3 = 144\).

Step 2: Illustrate Specific Orders

A meal order is a selection from each group. As an example, one such order could be: Ham, Potatoes, Coffee, and Cake. A second example might be: Fish, Green Beans, Milk, and Ice Cream. These are just two out of the 144 possible meal combinations.

Key Concepts

Understanding Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arranging, and combining objects. It plays a crucial role in determining the number of ways we can choose or arrange items from a set without the need to enumerate each possible choice manually.
Commonly used principles in combinatorics include permutations and combinations, which help to determine potential arrangements or selections based on different criteria.
The counting principle is another combinatorial tool, allowing us to calculate the total number of outcomes by multiplying the number of ways each individual event can occur.
For example, when determining how many different meals can be created from a restaurant menu with multiple categories, combinatorics helps us to efficiently find all possible combinations.

Creating Meal Combinations

Meal combinations involve selecting one item from each category—in this case, a main course, vegetable, beverage, and dessert. Considering all possibilities within these categories ensures no selection is left out.
The restaurant's menu offers:

• 4 main courses: Ham, Chicken, Fish, Beef
• 3 vegetables: Potatoes, Peas, Green Beans
• 4 beverages: Coffee, Tea, Milk, Soda
• 3 desserts: Cake, Pie, Ice Cream

To form a meal combination, choose one item from each list. For instance, one possible meal is "Chicken, Peas, Tea, Pie."
Another might be "Fish, Green Beans, Milk, Ice Cream." Each combination is unique and contributes to the total number of possible meals.

The Multiplication Rule in Counting

The multiplication rule is a fundamental principle in combinatorics used to calculate the total number of possible outcomes. This rule states that if there are multiple stages or categories in a decision-making process, the total number of outcomes is the product of the number of options available at each stage.
In the context of the restaurant menu, the multiplication rule can be applied as follows:

• Main Course: 4 choices
• Vegetables: 3 choices
• Beverages: 4 choices
• Desserts: 3 choices

Using the multiplication rule, the total number of meal combinations is calculated by multiplying these numbers together: \(4 \times 3 \times 4 \times 3 = 144\)
This means there are 144 different ways to create a meal by selecting one item from each of the four categories. This principle greatly simplifies the counting process, especially when dealing with multiple categories or large sets.