Problem 19 - Discrete Mathematics with Applications

Question

There are five types of desserts available at a restaurant. Find the number of ways eight people can select them, if order does not matter.

Step-by-Step Solution

Answer

There are 56 ways for eight people to select five types of desserts without considering the order.

Step 1: Understand the Problem

This problem is asking us to find the number of ways eight people can choose five types of desserts, without considering the order in which the desserts are chosen.

Step 2: Set up your variables

Let's set up our variables using the values given in the problem: - Number of people we have: \(n = 8\) - Number of types of desserts: \(k = 5\)

Step 3: Apply the combinations formula

Now let's apply the combinations formula to find the number of ways people can select desserts: \[ C(n, k) = \frac{n!}{k!(n - k)!} \] Plugging our values into the formula: \[ C(8, 5) = \frac{8!}{(5!)(8 - 5)!} \]

Step 4: Calculate the factorial for each number

Before calculating the combinations, we need to find the factorial for each number involved: - \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,\!320\) - \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\) - \((8 - 5)! = 3! = 3 \times 2 \times 1 = 6\)

Step 5: Calculate the final answer

Now we can calculate the number of combinations using the formula and the factorials we calculated: \[ C(8, 5) = \frac{40,\!320}{(120)(6)} \] \[ C(8, 5) = \frac{40,\!320}{720} = 56 \] There are 56 ways for eight people to select five types of desserts without considering the order.

Key Concepts

Combinations

In the given problem, we are dealing with a concept called combinations. Combinations allow us to determine how many ways we can select a group of items from a larger set, where the order of selection does not matter. This is ideal when choosing items like desserts for a group, where the sequence isn't important—just who gets what does.

When working with combinations, we use the formula:

• \( C(n, k) = \frac{n!}{k!(n - k)!} \)

Here, \(n\) represents the total items to choose from, and \(k\) is the number we want to select. The '!' symbol refers to factorial, which we'll discuss further in the next section. For our problem, it helps determine how many groups can be formed and is crucial for accurate results.

Factorials

Factorials play an essential role in combinatorics, especially in calculating combinations and permutations. Factorials are denoted by an exclamation point (\(!\)) and involve multiplying a series of descending natural numbers.

For example:

• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

Factorials help simplify the computation of combinations. They condense repetitive multiplications into a shorthand notation that makes solving problems like the dessert selection much more manageable.

In the problem, we computed \(8!\), \(5!\), and \(3!\) to plug into our combinations formula. Doing the factorial calculations first provides the building blocks for the final steps in solving.

Permutations

Permutations, like combinations, involve selecting items from a larger set. However, unlike combinations, for permutations, the order does matter. This changes the way we calculate them significantly.

For instance, if you were to arrange three types of desserts in a particular order, permutations would be used, as each arrangement of desserts is distinct. The formula for permutations differs from combinations and is given by:

• \( P(n, k) = \frac{n!}{(n - k)!} \)

In our problem, permutations would be irrelevant since the order of dessert selection does not matter. Recognizing when to use permutations versus combinations is critical in combinatorics.

Counting Principles

Counting principles refer to the basic rules used to evaluate the total number of possible outcomes or arrangements in a situation. This concept lays the foundation for understanding permutations and combinations.

There are two main principles:

• The Addition Principle: When one event cannot happen at the same time as another, the total number of outcomes is the sum of each event's outcomes.


• The Multiplication Principle: If events are independent and occur in sequence, the total number of outcomes is the product of the outcomes of each event.

In the problem at hand, we relied on these principles to decide that combinations, not permutations, was the suitable method to use, as the selection of desserts is independent and the order doesn't matter. Understanding these principles helps in setting up problem solutions correctly.