Problem 70 - A Graphical Approach to College Algebra

Question

Use any or all of the methods described in this section to solve each problem. Combination Lock A typical combination for a padlock consists of 3 numbers from 0 to \(39 .\) Count the number of combinations that are possible with this type of lock if a number may be repeated.

Step-by-Step Solution

Answer

There are 64,000 possible combinations.

Step 1: Understand the Problem

We need to determine the total number of combinations possible for a lock. Each combination consists of 3 numbers and each number is from 0 to 39.

Step 2: Identify the Range of Each Number

The lock has 40 numbers (from 0 to 39), meaning each position in the combination can be any of these 40 numbers.

Step 3: Determine if Repetition is Allowed

The problem states that a number may be repeated, meaning each position in the combination operates independently and numbers can repeat across the positions.

Step 4: Calculate Total Combinations

Since each of the 3 positions can be filled by any of the 40 numbers, the total number of combinations is calculated by multiplying the number of choices for each position: \[ 40 \times 40 \times 40 \]

Step 5: Perform the Multiplicative Calculation

Calculate the total number of combinations:\[ 40 \times 40 \times 40 = 40^3 = 64000 \]

Key Concepts

Permutations and Combinations

In combinatorics, understanding how to count arrangements and selections is key. Permutations deal with the arrangement of items where the order is important, whereas combinations focus on the selection of items where the order does not matter. However, when discussing a combination lock, the term "combination" refers to a specific sequence of numbers, not the mathematical concept of a combination.

• For a permutation, remember that the sequence a, b, c is different from c, b, a.
• For a combination in mathematics, both sequences a, b, c and c, a, b are considered the same.

In the context of a lock, we use the principles of permutations since the sequence matters; entering the code 1-2-3 is different from 3-2-1. This understanding helps clarify why we calculate the lock as a sequence problem, even though it's called a "combination lock."

Repetition in Combinations

Repetition is an important factor that influences how we count possibilities in a combination lock. The allowance of repetition means that numbers can be reused in different positions of a lock sequence.

• For instance, a number 5 could appear in all three positions: 5-5-5.

This changes the way we calculate combinations. If repetition were not allowed, each position would have fewer options as numbers are used, leading to a permutation without repetition. However, since repetition is allowed in the exercise, each position always has the same number of choices. This is mathematically approached as calculating single selections independently for each position and then multiplying the outcomes. This independence simplifies the computation of possibilities exponentially compared to when repetition is restricted.

Multiplicative Principle

The multiplicative principle is a foundational concept in combinatorics that allows us to easily compute the total number of outcomes. When each step in a process offers several independent choices, the multiplicative principle states that the total number of outcomes is the product of the number of choices for each step.

• In the combination lock example, each of the 3 digits represents a separate choice, each having 40 options.
• Hence, you multiply the possibilities: \[ 40 \times 40 \times 40 = 40^3 \].

This explains how such a large number of combinations—64,000 in this problem— is possible even with a seemingly limited range of numbers. This principle ensures that possibilities are properly counted, accounting for the independent selection at each stage.