Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Use any or all of the methods described in this section to solve each problem. The code for some garage door openers consists of 12 electrical switches that can be set to either 0 or 1 by the owner. With this type of opener, how many codes are possible? (Source: Promax.)

Step-by-Step Solution

Answer

There are 4096 possible codes.

Step 1: Understanding the Problem

The problem involves finding the total number of possible codes using 12 switches. Each switch can be set to either a 0 or a 1.

Step 2: Determine the Options for a Single Switch

For each switch in the garage door opener, there are exactly 2 options: 0 or 1. This means each switch can independently have one of two states.

Step 3: Calculate Total Combinations

Given that each of the 12 switches can have 2 states, the total number of combinations possible can be calculated by multiplying the number of possibilities for each switch: \[ 2^{12} \]

Step 4: Compute the Final Result

Evaluate the expression \(2^{12}\) to find the total number of possible configurations for the switches. \[ 2^{12} = 4096 \] This means there are 4096 possible codes.

Key Concepts

Binary Choices

When we talk about binary choices, we refer to situations where there are only two possible outcomes. In the context of this exercise, each switch on the garage door opener is a binary choice. It can only be set to either "0" or "1".

Binary choices are quite common in computing and digital systems. This is because computers operate using binary code, a system of "on" and "off" states. These states are represented as "1" and "0" respectively.

Understanding binary choices is crucial because it forms the foundation for computing systems and data representation. In this problem, each of the 12 switches represents a binary choice, creating a straightforward but powerful setup to calculate possible code combinations.

Exponentiation

Exponentiation is a mathematical operation involving two numbers. Here, it's used to determine the number of possible configurations for the switches. In our scenario, we have 12 switches, each of which can have 2 states (0 or 1).

Given these choices, the total number of combinations can be found using exponentiation. Specifically, we raise the number 2 (representing the two states of each switch) to the power of 12 (the number of switches):

• The base (2) signifies each binary option.
• The exponent (12) signifies the total number of switches.

The expression to calculate this is \((2^{12})\). Calculating this value will give us the number of potential unique codes that can be formed. In this exercise, \((2^{12} = 4096)\), meaning there are 4096 possible unique codes.

Problem-Solving Steps

Breaking down a problem into systematic steps is an effective problem-solving technique. Let's look at the steps used to solve the code configuration problem:

1. **Understanding the Problem**: Recognize what needs to be solved. Here, the goal was to find the total number of possible code combinations.
2. **Analyze Options**: Identify the options available - each switch has two options.
3. **Calculate Combinations**: Use mathematical tools like exponentiation to calculate the total number of combinations, based on the identified options.
4. **Evaluate Results**: Compute the expression to derive the answer.

Each step builds on the previous one, ensuring a comprehensive understanding and solution to the problem. By following structured steps, we simplify complex problems into manageable parts, making the solution more accessible.