Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Use any or all of the methods described in this section to solve each problem. A typical combination for a padlock consists of 3 numbers from 0 to 39 . Find the number of combinations that are possible with this type of lock if a number may be repeated. (Hint: The word combination is a misnomer. Lock combinations are permutations because the arrangement of the numbers is important.)

Step-by-Step Solution

Answer

There are 64,000 different sequences possible for the padlock.

Step 1: Understanding the Problem

We have a padlock that requires a three-number sequence from 0 to 39 to open it. We need to find out how many different sequences (or permutations) are possible if numbers can be repeated.

Step 2: Determining the Options per Slot

Each position in the sequence (first, second, and third) can be filled by any one of the numbers from 0 to 39. Therefore, there are 40 possible choices for each slot in the sequence. Repeated numbers are allowed.

Step 3: Calculating Total Permutations

Since there are 40 options for each of the 3 positions in the lock sequence, we calculate the total number of permutations by multiplying the number of choices for each slot together: \[ 40 \times 40 \times 40 = 40^3 \]

Step 4: Compute the Total

Calculate \( 40^3 \) to find the total number of permutations.\[ 40^3 = 64000 \]

Step 5: Conclusion

Thus, there are 64,000 different possible permutations for the padlock sequence.

Key Concepts

Combinatorial Mathematics

Combinatorial Mathematics is a field of mathematics focused on counting, arranging, and analyzing arrangements of objects. In this context, we explore permutations, a type of arrangement where the order of objects matters.
For the padlock problem, each sequence of numbers represents a different arrangement. Unlike combinations, where order does not matter, permutations emphasize arrangement importance. This distinction is crucial because modifying the order of numbers in a padlock changes the sequence entirely, leading to a different permutation.

• Permutations consider the order relevant;
• Repeated elements are allowed;
• Each slot is independent in choice.

Using the principle of multiplication, we can calculate the total possible permutations. When a number can be repeated, as in our exercise, the number of choices for each position remains constant. Therefore, the calculation becomes straightforward, involving the product of the number of choices for each position.

Discrete Mathematics

Discrete Mathematics deals with distinct and separate values, often integers, rather than continuous. It covers topics such as logic, set theory, and graph theory, focusing on countable structures.
In our lock example, each number sequence for the padlock is a discrete set of values. The numbers range from 0 to 39, giving us 40 distinct options for each place in the permutation.
This scenario excludes any form of continuity and emphasizes individual choice slots. Understanding discrete mathematics helps students individually consider each possible number in the sequence and compute their total arrangements logically.
Instead of thinking about numbers as continuous intervals, like real numbers, we treat them as finite distinct objects.

• Each number in the sequence is distinct.
• The focus is on integers and finite choices.

Mathematical Reasoning

Mathematical Reasoning is a vital skill in problem-solving, involving both logical deduction and the application of mathematical concepts. When solving permutation problems, it's key to apply systematic reasoning to break down complex tasks into simpler, manageable steps.
For instance, in our exercise:

• Identify each slot in the padlock as a separate choice.
• Recognize the importance of sequence order in permutations.
• Assess the problem constraints, like repetition.

Engage mathematical reasoning to recognize that repetition permits 40 possible numbers for each of the three slots. The permutation calculation involves multiplying these choices to reflect each independent decision for the sequence. Applying these reasoning techniques simplifies understanding complex problems. It enhances strategic planning, ensuring every step aligns with the solution's logical framework.