Problem 38 - A Graphical Approach to College Algebra

Question

Use the fundamental principle of counting or permutations to solve each problem. Meal Choices A menu offers a choice of 3 salads, 8 main dishes, and 5 desserts. How many different 3-course meals (salad, main dish, dessert) are possible?

Step-by-Step Solution

Answer

There are 120 different 3-course meal combinations possible.

Step 1: Identify Each Course Category

You have three categories of courses: salads, main dishes, and desserts. Each offers different options.

Step 2: Determine Choices for Each Course

The problem states there are 3 salad options, 8 main dish options, and 5 dessert options.

Step 3: Apply the Fundamental Principle of Counting

Multiply the number of choices for each course to find the total number of combinations. That is, multiply the number of salads, main dishes, and desserts: \(3 \times 8 \times 5\).

Step 4: Calculate the Total Number of Meals

Perform the multiplication: \(3 \times 8 \times 5 = 120\). Hence, there are 120 different combinations of meals possible.

Key Concepts

Fundamental Principle of Counting

Imagine you are at a restaurant with a delicious menu. The Fundamental Principle of Counting helps you determine how many different types of meals you could create. This is a key concept in combinatorics, focusing on how counting different combinations of choices is done. For example, consider you want a meal with a salad, a main dish, and a dessert. Each category has a list of options, and you need to know the number of possible meal combinations.

For each salad option you pick, there are several main dishes and desserts you can choose from. To figure out the number of possible meals, simply multiply the number of choices in each category. Let's break it down further:

• If there are 3 salads, for each salad option, you can choose from a number of main dishes.
• For each main dish, pair it with any of the desserts.

Thus, using the Fundamental Principle of Counting, by multiplying the choices in each category, you find the total combinations. It's like adding layers to your meal choice, ensuring you account for every possible outcome.

Permutations

Permutations are another important topic in the wider concept of combinatorics. However, when it comes to deciding meal combinations, using permutations might not always be applicable. Permutations are focused on the arrangement of items, considering the order significant.

If, for instance, each different order of eating the meal (e.g., salad first, then dessert, followed by a main dish) mattered, then permutations would be relevant. However, since we are considering just the selection of distinct items without regard to the order in this scenario, permutations aren't directly involved. Instead, permutations are more useful in problems where you need to arrange or sequence items rather than simply select them from set categories.

Multiplication Principle

The Multiplication Principle is a cornerstone in solving counting problems like the meal choice scenario. It simplifies finding combinations by multiplying the number of choices across separate categories, often seen in problems involving distinct steps or parts.

Here’s how it works: if you have multiple stages or decisions, and each stage has several options, multiplying these options gives you the total count of ways to complete all stages. For meals:

• Step 1: Choose a salad. You have 3 choices.
• Step 2: For each salad, choose among 8 main dishes.
• Step 3: For each salad and main course pairing, pick from 5 desserts.

So, when you multiply the number of salads, main dishes, and desserts together (3 x 8 x 5), you get the total number of meal combinations. The Multiplication Principle is a simple yet powerful tool for these kinds of calculations in combinatorics, making seemingly complex problems quite manageable.