Problem 18 - College Algebra with Modeling and Visualization

Question

\(\mathbf{A}\) typical combination for a padlock consists of three numbers from 0 to 39. Count the combinations possible with this type of lock if a number may be repeated.

Step-by-Step Solution

Answer

There are 64,000 possible combinations for the padlock.

Step 1: Understanding the Problem

We need to determine the number of possible combinations for a lock, where each combination consists of three numbers, ranging from 0 to 39, and numbers can be repeated.

Step 1: Identify the Range

The padlock uses numbers from 0 to 39. This indicates there are 40 possible choices for each number in the combination (since counting from 0 to 39 includes 40 numbers).

Step 2: Apply the Counting Principle

For each position (first, second, and third) in the three-number combination, there are 40 possible choices. The total number of combinations is the product of choices for each position.

Step 3: Calculate the Total Combinations

Since there are 40 choices per position and 3 positions, use the formula for combinations with repetition: \( 40 \times 40 \times 40 = 40^3 \).

Step 4: Compute the Result

Calculate the result of the expression: \( 40^3 = 64,000 \).

Key Concepts

Counting Principle

The counting principle is a fundamental concept in combinatorics. It helps us determine the number of possible outcomes when there are multiple events happening consecutively. To use it, you simply multiply the number of ways each event can occur.

For example, if you have three boxes and each box can be filled with either a red, blue, or green ball, the counting principle tells you to multiply the number of choices for each box: 3 choices for the first box, 3 for the second, and 3 for the third, resulting in a total of 3 x 3 x 3 = 27 possible combinations.

In the context of the padlock combination problem, we have 3 positions to fill (similar to the three boxes) and each position can have any number from 0 to 39. Hence, there are 40 choices for each position. Using the counting principle here, the possible combinations are calculated by multiplying: 40 for the first digit, 40 for the second, and 40 for the third, yielding a total of 40 x 40 x 40 = 64,000 combinations.

Combinations with Repetition

In many situations, especially like the padlock problem, combinations with repetition come into play. This is when you have to consider repeating numbers or elements in a sequence.

• Imagine having an unlimited supply of 0-39 to use three times for the lock.
• Repetition means you can use the same number multiple times in different positions.
• In this kind of scenario, repetition allows for more combinations since each element can be reused.

Unlike permutations, where order makes each combination distinct, combinations with repetition focus simply on the selection of elements, including repeated elements.

In our padlock example, we are dealing with combinations with repetition because numbers can repeat. Since each position in the combination can independently be any of the 40 possible numbers, you're effectively selecting with the freedom for repetition. Therefore, our calculation boils down to 40 options raised to the power of 3 (one for each position), resulting in 40^3 = 64,000 possible combinations.

Mathematical Problem-Solving

Mathematical problem-solving is an essential skill that involves understanding a problem, planning an approach, carrying out a plan, and finally reviewing the result.

• The first step is comprehension — grasping what the problem is asking. For our padlock problem, this means realizing it involves repetitive combinations of numbers.
• Then, devise a plan. Here, using the counting principle helps structure the calculations.
• Implementation involves executing your plan. We calculated the number of combinations using multiplication.
• Finally, always revisit the problem to ensure the solution makes sense. Calculating 40^3 and obtaining 64,000 should solve our question convincingly.

By following a structured approach, we not only solve the mathematical problem of padlock combinations but also sharpen our skills in reasoning and logical thinking. Each step needs careful consideration and teaches us how to break down complex problems into manageable tasks.