Problem 24 - College Algebra

Question

Use the Fundamental Counting Principle to solve Exercises \(21-32\). A restaurant offers the following lunch menu. Main Course Vegetables \(\quad\) Beverages \(\quad\) Desserts \begin{tabular}{llll} \hline Ham & Potatoes & Coffee & Cake \\ Chicken & Peas & Tea & Pie \\ Fish & Green beans & Milk & Ice cream \\ Beef & & Soda & \end{tabular} 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

In total 144 meal combinations can be ordered from the menu.

Step 1: Identify The Number of Choices per Category

There are four categories of items on the menu: main course, vegetables, beverages, and desserts. The number of choices in each category is as follows - Main Course: 4 (Ham, Chicken, Fish, Beef), Vegetables: 3 (Potatoes, Peas, Green beans), Beverages: 4 (Coffee, Tea, Milk, Soda), Desserts: 3 (Cake, Pie, Ice Cream)

Step 2: Apply the Fundamental Counting Principle

The Fundamental Counting Principle states that if there are n ways to do one thing, and m ways to do another, then there are n*m ways of doing both. Therefore, the total number of different meals that can be ordered is given by multiplying the number of choices in each category. So, the total number of possible orders is \(4 (Main Course choices) * 3 (Vegetable choices) * 4 (Beverage choices) * 3 (Dessert choices) = 144\).

Step 3: Describe two such orders

Two possible orders could be: 1) Ham with Potatoes, served with Coffee, and Cake as Dessert. 2) Beef with Green beans, served with Soda, and Ice Cream as Dessert.

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination of objects. It helps answer questions like how many ways certain events can occur. In the given exercise, combinatorics helps determine the number of possible meal combinations from a restaurant menu.

By understanding the choices available and how they can be combined, we use combinatorics to find solutions efficiently. This involves applying tools like the Fundamental Counting Principle, which simplifies the process of calculating the total number of possible combinations by considering the product of the individual choices.

When applying combinatorics, especially in real-world problems like the one with the restaurant menu, it's essential to identify each category (e.g., main course, beverages, etc.) and count the choices in each. Then, multiply the number of options across categories to find the total combinations.

Probability

Probability is the measure of the likelihood that an event will occur. In this exercise's context, it's about calculating how likely it is to randomly order any specific meal from the menu.

When we know the total number of meal combinations, which is 144, thanks to our combinatorial analysis, the probability of picking a specific meal depends on how many occurrences of that meal exist among those combinations.

For instance, if someone has their heart set on Ham with Potatoes, served with Coffee and Cake, the probability would be calculated as 1 (for the single specific combination we desire) divided by 144 (the total possible combinations). This gives a very small probability of approximately 0.0069, or 0.69% chance of selecting that exact meal at random.

Menu Combinations

Menu combinations refer to the different ways items can be selected from each category to form a complete meal. In the exercise, category options include main courses, vegetables, beverages, and desserts.

Each choice in these categories multiplies with choices from the others to form a variety of meals. We calculated 144 possible combinations by multiplying the number of options: 4 (Main Courses) × 3 (Vegetables) × 4 (Beverages) × 3 (Desserts).

When considering menu combinations, think about the variety you can create. For instance, one could choose Ham with Potatoes and Coffee, followed by Cake, or switch things around entirely. Menu combinations highlight flexibility and diversity, which are key to understanding outcomes in such combinatorial problems.

Permutations

Permutations involve arrangements where the order does matter. Unlike our exercise with menu combinations where selection order doesn't change the outcome (choosing Ham then Potatoes is the same as choosing Potatoes then Ham, essentially), permutations must consider the sequence.

Although permutations are not directly applied in our restaurant menu problem, understanding this concept elevates your grasp on more complex scenarios where order is crucial. For instance, if items must be prepared or served in a particular sequence, permutations would be the tool to count those scenarios.

Basic comprehension of permutations can also aid in solving problems where arranging a sequence of actions or events is needed, emphasizing not just selection but the order of these selections.