Question

A French restaurant in Riverside, California, offers a special summer menu in which, for a fixed dinner cost, you can choose from one of two salads, one of two entrees, and one of two desserts. How many different dinners are available?

Short answer

Answer: There are 8 different dinners available on the summer menu.

Step-by-step solution

Identify the number of choices for each category

We are given that there are two choices for each category (salad, entree, and dessert). So, number of salads = 2 number of entrees = 2 number of desserts = 2

Apply the counting principle

Since there are 2 choices for each of the 3 categories, we must find the product of all these choices to find the number of possible dinners. According to the counting principle, the number of possible dinners is: Number of dinners = Number of salads * Number of entrees * Number of desserts

Calculate the number of dinners

Using the values from step 1, we can now calculate the number of different dinners: Number of dinners = 2 * 2 * 2 = 8 Thus, there are 8 different dinners available on the summer menu at the French restaurant in Riverside, California.

Key concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and finding patterns within sets of objects. It plays a crucial role in solving problems where you need to determine the number of possible outcomes, like in our restaurant menu example.
To understand this, consider a menu where you're asked to choose one dish from each category (salad, entree, dessert). Combinatorics helps us determine how many meal combinations are possible by using something called the "Counting Principle." This principle states that if there are "n" ways to do one thing and "m" ways to do another, there are \(n \times m\) ways to do both.

• If the menu has 2 salads, 2 entrees, and 2 desserts, each category offers 2 choices.
• To find the total combinations, multiply the choices in each category: \(2 \times 2 \times 2 = 8\) possible dinners.

Focus on making sensible combinations, and remember that each choice in one category doesn't affect the other categories. This organized approach highlights the beauty of combinatorics in simplifying and solving such real-world problems.

Probability

Probability revolves around understanding the likelihood of different outcomes. It helps us reason about how likely certain combinations or events are to occur. Although calculating the total number of combinations, like with the dinner menu, doesn't directly involve probability, the concept is closely related.
In scenarios where you need to consider the likelihood of each meal option being chosen, probability becomes essential. Assuming each dish is equally likely to be picked, you can determine the probability of selecting a specific meal combination. This notion becomes especially handy when predicting or analyzing preferences and can aid in broader applications like planning or menu adjustments.

• Probability of selecting any specific combination: Given all combinations are equally likely, the probability is \(\frac{1}{8}\).
• Probabilities help estimate outcomes and inform decisions, whether in dining or daily life.

The simplicity of counting combinations lays a foundation for assessing probabilities, allowing restaurants or planners to evaluate the probable popularity of different menu items.

Mathematical Calculation

Mathematical Calculation is an essential part of tackling combinatorial and probability problems. It involves performing operations like multiplication, as seen when calculating the total number of menu options in our example.
By breaking down the problem's specifics, such as identifying the fixed choices in categories, mathematical calculations serve to provide clear and direct answers.

• In our exercise, by knowing there are 2 salads, 2 entrees, and 2 desserts, use multiplication: \(2 \times 2 \times 2 = 8\).
• This straightforward method ensures efficiency and accuracy, minimizing errors in arithmetic operations.

Math calculations facilitate quick solutions to even complex issues, highlighting their importance across various real-world contexts. Make sure to align your math operations with the logic of the problem for reliable results.