Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Lunch possibilities Owners of a restaurant advertise that they offer \(1,114,095\) different lunches based on the fact that they have 16 "free fixins" to go along with any of their 17 menu items (sandwiches, hot dogs, and salads). How did they arrive at that number?

Step-by-Step Solution

Answer

The number 1,114,095 comes from combining each of the 16 fixins in \(2^{16}\) ways with 17 menu items.

Step 1: Determine Fixins Combinations

To determine the number of lunch combinations using the fixins, we need to consider each unique combination of the 16 fixins. Each fixin can either be included or not in a lunch, which gives us two possibilities (include or not) for each fixin. This means the total number of combinations of fixins is calculated as \(2^{16}\).

Step 2: Calculate Total Lunch Combinations

Given the 17 menu items, each combination of fixins can be paired with any of these menu items. Therefore, to find the total number of lunch combinations, multiply the total number of fixins combinations by the number of menu items: \(2^{16} \times 17\).

Step 3: Simplify the Mathematical Expression

Calculate the number of fixins combinations: \(2^{16} = 65536\). Then, multiply by the number of menu items: \(65536 \times 17 = 1,114,095\). This confirms the number of different lunches advertised.

Key Concepts

Combinations

Combinations are fundamental in calculating the number of possible variations in a group of items. Think of combinations as the different ways you can choose items from a collection where the order doesn't matter. In our lunch scenario, we calculate combinations to determine how we can use the 16 fixins. For every fixin, we have two options: include it in the lunch or not. This situation is classic combination math as each decision is independent of the others.
To fathom the number of ways we can combine these fixins, use the formula of combinations with repetition, given as\(2^n\). Here, \(n\) is the number of fixins. Each choice is binary (yes or no), thus there are two different ways for each fixin. Given 16 fixins, the complete calculation becomes \(2^{16}\), illustrating why combinations are key in such problems.

Exponents

Exponents are used in combinatorics to calculate possibilities for binary choices quickly. An exponent features a base number and a power, written like \(a^b\). It represents multiplying the base by itself as many times as indicated by the exponent. In the restaurant's situation, each of the 16 fixins can either be included or excluded. We can represent this choice as \(2\) (the base), raised to the power of \(16\) (the number of fixins), thus \(2^{16}\).
When expanding it:

• For 1 fixin: \(2^1 = 2\) choices
• For 2 fixins: \(2^2 = 4\) choices
• ...and so on until...
• For 16 fixins: \(2^{16} = 65536\) choices

This pattern showcases how exponentially growth captures the vast array of combinations possible with multiple items.

Multiplication

Multiplication in this context helps us find the total number of outcomes by combining different elements. We use multiplication to calculate possibilities when multiple independent events or decisions combine to form a single outcome. After determining the combinations of fixins as \(65536\), we multiply this by the number of menu items, which is 17.
This multiplication step:

• \(65536\) (ways to select fixins)
• \(\times 17\) (different menu items)

Gives a total of \(1,114,095\) ways to put together a lunch. Multiplication assists in combining possibilities across different independent choices, showing its importance in such modeling tasks.

Mathematical Modeling

Mathematical modeling is a way to describe real-world problems in mathematical language, helping us solve them using mathematical operations. In our restaurant problem, mathematical modeling involves creating a formula that encapsulates lunch combinations. It's dynamic, depicting how varied inputs like numbers of menu items and fixins impact outcomes.
This model:

• Starts with understanding the decision points: Selecting fixins and a menu item.
• Utilizes exponents to handle the binary fixins choices, simplifying multiplicity.
• Employs multiplication to combine fixins combinations with the menu options.

Each step in the model systematically mirrors the decisions and leads to the final calculation of possibilities. Such modeling enables us to predict and compute possibilities efficiently in complex scenarios.