Problem 62 - Algebra and Trigonometry

Question

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

There are 1,048,576 different possible outcomes.

Step 1: Understand the Problem

There are 20 shoppers, each making a binary choice: to enter the store or not. We need to calculate the number of different possible outcomes of their decisions.

Step 2: Determine Outcomes for One Shopper

Each shopper has 2 possible decisions: enter the store or do not enter the store.

Step 3: Calculate Total Outcomes for All Shoppers

Since each of the 20 shoppers makes an independent decision, the total number of outcomes is given by multiplying the number of choices for each shopper. This is calculated as \(2^{20}\).

Step 4: Simplify the Calculation

Calculate \(2^{20}\) to find the total number of different outcomes. \(2^{20} = 1,048,576\).

Key Concepts

Binary Choices

In the given scenario, each shopper faces a binary choice: to either enter the clothing store or not. Binary choices are decisions that have only two options. They are like a fork in the road where you can only take one path. Here are some examples of binary choices:

• Yes or No
• On or Off
• Enter or Do Not Enter

Binary choices are fundamental in combinatorial problems as they lay the groundwork for calculating possible outcomes. It's important to understand that even with a simple decision set, like entering a store or not, the complexity can increase dramatically when the number of participants increases.

Independent Decisions

Each shopper's decision in the shopping mall is independent of the others. Independent decisions mean that the choice made by one shopper does not affect the choices available to others. This aspect is crucial because it allows us to use a simple multiplication rule to determine the total number of outcomes.
Consider two independent events, each with its own set of outcomes. If one shopper decides whether to enter or not, that decision is not influenced by the decision of another shopper. Thus, each decision stands alone, making it possible to calculate total outcomes by multiplying the number of independent choices for each participant.

Power of Two

Understanding "power of two" is key to solving this kind of combinatorial problem. When each participant has two independent choices, like entering or not entering, and each choice is independent of others, the problem becomes a matter of calculating powers of two.
For 20 shoppers each having two choices, the formula to calculate the total outcomes is given by using the exponent on 2, which signifies each independent choice. This is expressed as: \[ 2^{20} \]Such powers of two calculations arise frequently when addressing problems involving binary decisions. Powers of two grow exponentially, which means even a small increase in the number of people can lead to a significant increase in the number of possible outcomes.

Decision Outcomes

The outcome of this exercise tells us the total number of ways in which the 20 shoppers can make their choices. This is calculated as \(2^{20}\), which equals 1,048,576.
These decision outcomes demonstrate the power and complexity involved when even a small group of people makes simple binary decisions independently. Understanding how to calculate these different decision outcomes enables predictions and planning for various scenarios in real-world situations, like determining potential customer flow in a shopping mall.
Each decision pathway one shopper chooses creates a unique result, and when 20 independent decisions are made, the combination of these decisions leads to a large number of possible unique outcomes.