Problem 26 - College Algebra with Modeling and Visualization

Question

A menu offers 5 different salads, 10 different entrees, and 4 different desserts. How many ways are there to order a salad, an entrée, and a dessert?

Step-by-Step Solution

Answer

There are 200 ways to order a salad, entrée, and dessert.

Step 1: Understand the Problem

We need to find the total number of ways to choose one salad, one entrée, and one dessert from the menu.

Step 2: Identify the Options for Each Course

The menu provides 5 different salads, 10 different entrees, and 4 different desserts to choose from.

Step 3: Apply the Fundamental Counting Principle

According to the Fundamental Counting Principle, if there are \( m \) ways to do one thing and \( n \) ways to do another, then there are \( m \times n \) ways to do both. In this case, the number of ways to select a complete meal consisting of a salad, an entrée, and a dessert is given by multiplying the number of options for each course.

Step 4: Perform the Multiplication

Calculate the total number of ways by multiplying the options for each course: \( 5 \) salads, \( 10 \) entrées, and \( 4 \) desserts, so the total ways are \( 5 \times 10 \times 4 \).

Step 5: Calculate the Result

Complete the multiplication: \( 5 \times 10 = 50 \), and then \( 50 \times 4 = 200 \). Therefore, there are 200 different ways to select one salad, one entrée, and one dessert.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arrangement, and combination of objects. It deals with understanding the various ways we can arrange and select items from a group without overlooking any possibilities.
In this exercise about a menu selection, combinatorics helps us solve how to arrange our meal by choosing one item from each category—salads, entrees, and desserts. The principle used here is a fundamental aspect of combinatorics which aids in determining the total number of ways to choose items.

• The crux of combinatorics in this scenario is determining all possible combinations of one salad, one entrée, and one dessert.
• Each selection does not affect the others, allowing the combination of these independent choices to form complete meals.

By breaking down the decision-making process through combinatorics, we can easily count and verify the possible outcomes, ensuring no combination is missed. Understanding this core concept is essential in tackling similar complex mathematical problems involving counting and arrangements.

Problem-Solving in Mathematics

Mathematical problem-solving is a skill essential for navigating through exercises like the one about choosing meals. It involves systematically approaching a problem and breaking it down into manageable steps.
In the given exercise, the first step is identifying the problem: finding the total number of ways to make a full meal by choosing one item from each category. Recognizing this primary objective gives clarity and direction for finding the solution.

• Breaking down the choices into separate groups (salads, entrees, and desserts) simplifies the problem-solving process.
• Applying relevant mathematical principles, like the Fundamental Counting Principle, helps consolidate understanding and find the solution.

Problem-solving is not just about calculation; it's about clear-thinking, structured approaches, and systematic methods that assist in overcoming challenges and complex questions efficiently.

Multiplication in Algebra

Multiplication in algebra is not just about computing numbers; it extends into interpreting mathematical problems and solutions, as demonstrated in this exercise. The multiplication is applied when multiple independent choices need to be combined to find a total outcome.
In our meal selection scenario, the use of multiplication occurs because we are combining different independent groups of choices:

• There are 5 salads, 10 entrees, and 4 desserts available.
• By multiplying these numbers, we calculate all the possible combinations of these independent selections.

The algebraic principle allowing for such calculations is simple multiplication, expressing the "and" condition in choices—"select one salad and one entrée and one dessert."
This application of multiplication simplifies the process of counting possibilities and is a fundamental skill in algebra, often employed in solving real-world mathematical problems.