Problem 60 - Algebra and Trigonometry

Question

55–75 Solve the problem using the appropriate counting principle(s). To Shop or Not to Shop Each of 20 shoppers in a shopping mall chooses to enter or not to enter the Dressfastic clothing store. How many different outcomes of their decisions are possible?

Step-by-Step Solution

Answer

1,048,576 different outcomes are possible.

Step 1: Understanding the Problem

We need to determine how many different outcomes are possible when each of 20 shoppers decides either to enter or not to enter a store.

Step 2: Analyzing One Shopper's Decision

Each shopper has exactly 2 choices: either to enter the store or not to enter the store. This situation can be represented as having 2 possible outcomes for every shopper.

Step 3: Applying the Fundamental Counting Principle

According to the fundamental counting principle, the total number of outcomes for a sequence of independent events is the product of the number of outcomes for each event. Here, we have 20 independent shopper decisions.

Step 4: Calculating Total Outcomes

Since each decision has 2 possible outcomes and there are 20 independent shoppers, we use the formula for the counting principle: \[ 2^{20} \]This represents the total number of different outcomes possible.

Step 5: Computing the Power

Now we calculate the value of \(2^{20}\). This gives us the total number of different combinations of decisions the 20 shoppers can make.

Step 6: Final Result

Calculating \(2^{20}\) yields the total number of outcomes: 1,048,576.

Key Concepts

Fundamental Counting Principle

The fundamental counting principle is a key concept in combinatorics and probability theory. It helps us determine the total number of possible outcomes when there are multiple independent choices or events to consider. To understand this principle, consider a simple scenario: if you have two independent events where each can happen in different ways, then the total number of outcomes is the product of the number of choices for each event.
In the shopping mall problem, each shopper's decision represents an independent event. Each shopper can choose either to enter or not enter the store, giving 2 possible outcomes per shopper. According to the fundamental counting principle, if you multiply the number of choices each shopper has, this provides the total number of possible combinations for all shoppers. By extending this logic to 20 shoppers, each having 2 choices, we use the multiplication rule to find the total:

• Number of outcomes per shopper = 2
• Total number of shoppers = 20
• Total possible outcomes =\(2 \times 2 \times ... \text{(20 times)} = 2^{20}\)

Independent Events

Understanding independent events is crucial to applying the fundamental counting principle correctly. Independent events are those whose outcomes do not affect one another. In other words, the occurrence or result of one event does not change the likelihood of the occurrence of another event.
For example, in our scenario, whether one shopper decides to enter the store or not doesn't influence the decision of another shopper. Each shopper makes a decision independently. This independence is what allows us to simply multiply the number of choices each shopper has without needing adjustments for interdependencies.
In mathematical terms, if Event A and Event B are independent, the probability of both events occurring is the product of their individual probabilities. This concept extends to counting, where the number of outcomes for several independent events is the product of the number of outcomes for each individual event. For our 20 shoppers, each making an independent decision, this independence allows us to use the expression \(2^{20}\) to signify all possible combinations of decisions.

Outcomes Calculation

Outcomes calculation is a straightforward application of the fundamental counting principle, particularly when dealing with independent events. In this exercise, the problem involves calculating the total number of outcomes for a scenario with multiple binary decisions—each decision being a shopper's choice to enter or not enter the store.

• The calculation using the fundamental counting principle involves multiplying the number of outcomes for each independent decision.
• For 20 shoppers, each with 2 options (enter/not enter), the total number of outcomes is calculated as \(2^{20}\).
• By evaluating \(2^{20}\), we derive the total number of possible outcomes: 1,048,576.

This large number indicates the many combinations in which the shoppers can make their decisions, providing comprehensive insight when planning for all possible scenarios in this context. Understanding outcomes calculation through such problems showcases the power of effective counting principles.