Question

A French restaurant 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 dinner combinations available.

Step-by-step solution

Identify the number of options for each course

In the restaurant's summer menu, there are: - 2 options for salads (A or B) - 2 options for entrees (C or D) - 2 options for desserts (E or F)

Apply the counting principle

According to the counting principle, the total number of dinner combinations can be found by multiplying the number of options for each course together. So, the calculations would be: Number of dinner combinations = (Number of salad options) × (Number of entree options) × (Number of dessert options)

Calculate the number of dinner combinations

Now, we will substitute the number of options for each course into the equation from Step 2: Number of dinner combinations = (2) × (2) × (2)

Compute the result

By multiplying the numbers together, we can determine the total number of available dinner combinations: Number of dinner combinations = 8 So, there are 8 different dinner combinations available on the French restaurant's special summer menu.

Key concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics that explores the art of counting and arranging objects. It's at the heart of many problems involving choices and arrangements and is incredibly useful in various fields, from computer science to everyday scenarios like choosing menus. When we talk about combinatorics, we are usually interested in determining how many different ways we can arrange or choose items.
One key aspect of combinatorics is identifying the distinct arrangements or selections possible in a given situation. This often leads to deeper questions like: How would the total number of combinations change if more options were added, or we had fewer elements to choose from? By understanding combinatorics, we can simplify complex problems into manageable ones and find solutions more systematically.

Multiplicative Rule

The multiplicative rule is a cornerstone concept in combinatorics that helps solve problems involving multiple independent choices. It states that if there are several stages of selection, and choices are made independently at each stage, the total number of outcomes is the product of the number of choices at each stage.
In the context of our summer menu exercise, the multiplicative rule shows its power. For the French restaurant example, you have:

• 2 choices for salads,
• 2 choices for entrees,
• 2 choices for desserts.

To find the total number of dinner combinations, you simply multiply these numbers:
\[2 \times 2 \times 2 = 8\]This rule helps in calculating possibilities efficiently, and it's a foundational tool for understanding how distinct outcomes can be formed when dealing with independent events.

Permutations and Combinations

While permutations and combinations may seem similar, they handle different types of problems in combinatorics. Both concepts are about arranging and selecting items but differ in whether order matters.

Permutations

Permutations refer to arrangements where order is significant. For example, if assigning seats in a shuttle, rearranging the passengers yields different permutations, because seating order matters. However, in our summer menu context, permutations don't apply because choosing salad A before dessert F is the same as choosing dessert F before salad A.

Combinations

Combinations involve selections where the order does not matter. Our restaurant menu is a prime example of using combinations—we only care about the different sets of food items, not the order in which they are selected. The total number of dinner combinations calculated in this exercise relied on the principle of combinations.
Understanding when to use permutations vs. combinations is crucial because each applies to different scenarios and impacts how we approach a counting problem.