Problem 38 - Algebra 2 and Trigonometry

Question

To duplicate a key, a locksmith begins with a dummy key that has several sections. The locksmith grinds a specific pattern into each section. a. A particular brand of house key includes 6 sections, and there are 4 possible patterns for each section. How many different house keys are possible? b. A desk key has 3 sections, and 64 different keys are possible. How many patterns are available for each section if each section has the same number of possible patterns?

Step-by-Step Solution

Answer

a. 4096 different house keys. b. 4 patterns per section.

Step 1: Understand the Problem

We have two problems to solve: finding the number of possible house keys for a key with 6 sections, each with 4 possible patterns; and determining how many patterns are possible per section of a desk key with 3 sections and 64 different possible keys.

Step 2: House Key Calculation

To find the number of possible house keys, multiply the number of patterns for each section (4) by itself for the number of sections (6). This is because each section can be any one of the 4 patterns, independently. The formula is: \( 4^6 \).

Step 3: Calculate Power

Calculate \( 4^6 \). Multiply 4 by itself six times:\[4^6 = 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4096.\]Therefore, there are 4,096 different possible house keys.

Step 4: Desk Key Calculation

For the desk key, you are given the total number of keys (64) and need to find the number of patterns per section when there are 3 sections. Use the formula \( n^3 = 64 \), where \( n \) is the number of patterns per section.

Step 5: Solve for n

Calculate the cube root of 64 to find \( n \):\[n = \sqrt[3]{64} = 4.\]This indicates there are 4 possible patterns per section for the desk key.

Key Concepts

Permutation

Permutation in combinatorics refers to the arrangement of a set of items in a particular order. However, in the context of key making, permutation can be viewed when each section of the key can have one of several distinct patterns.
Imagine you have a key with three sections, and each section can be one of 4 patterns: A, B, C, or D. A permutation for this key could be arranged like A-B-C, D-C-B, or any other combination where pattern order matters.
This concept is vital in understanding how keys differ according to unique arrangements of patterns. When a locksmith creates keys, each permutation of patterns generates a unique key.
Here are some key points regarding permutations in lock-making:

• Permutations can occur with independent choices for each section.
• Order of patterns affects the resulting key.
• The total permutations can be calculated in cases where each pattern is distinct across sections.

Through permutations, a locksmith can calculate and create a wide range of unique keys, enhancing security for locks.

Exponentiation

Exponentiation is a mathematical operation where a number, called the base, is multiplied by itself a certain number of times. This can be visualized as repeated multiplication.
In key making, to determine how many possible keys can be generated from sections, exponentiation plays a crucial role.
For a house key with six sections, each having 4 possible patterns, the total number of combinations is calculated as an exponentiation of the number of patterns per section raised to the power of the number of sections:

• Base = number of patterns per section (4).
• Exponent = number of sections (6).

Thus, the calculation is given by:
\[ 4^6 = 4096 \]
This means there are 4,096 possible combinations for the house key. This computation shows how exponentiation helps in evaluating the possibilities inherent in combinatorial problems.
Understanding exponentiation can help students see how powerful this mathematical tool is in the context of key combination possibilities.

Counting Principle

The Counting Principle is a fundamental concept in combinatorics used to determine the number of possible outcomes in a sequence of events or series of choices. It is particularly useful when there are multiple independent choices to be made.
In our key making scenario, the Counting Principle explains how to find the total number of different keys. For keys with several sections, where each section can independently have a certain number of patterns, the total is computed simply by multiplying the number of patterns for each section.
For a key with 6 sections and 4 possible patterns for each section:

• Calculate the number of outcomes for one section (= 4).
• Multiply these choices across all sections (= 6).

This approach enables calculation of:\[ 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4^6 = 4096 \]
Thus, using the Counting Principle, one can efficiently determine the total number of unique keys, underscoring its value in combinatorial situations.

Patterns in Key Making

Patterns in key making relate directly to the concept of unique sequences created by combining different patterns or grooves a key can have. Each section of a key can feature various patterns which determine its fit to a particular lock.
Locks and their keys rely on the distinct configuration of these patterns. Each key section can have multiple grooves symbolize multiple choices, and the correct sequence of sections with specific patterns makes the key functional.
When crafting keys, a locksmith needs to consider:

• The number of sections each key has.
• The number of different patterns possible for each section.
• Unique combinations of patterns that result in a functional key.

For example, in the given exercise, with a desk key having 3 sections and each capable of 4 unique patterns, the permutation of these would create numerous distinct keys. This helps ensure security and exclusivity for locks.
In essence, patterns create the foundational arrangements that every unique key depends on, allowing both physical security and logical organization of possibilities.