Problem 71 - A Graphical Approach to College Algebra

Question

Use any or all of the methods described in this section to solve each problem. Combination Lock \(\mathrm{A}\) briefcase has 2 locks. The combination to each lock consists of a 3 -digit number, where digits may be repeated. How many different ways are there of choosing the six digits required to open the briefcase?

Step-by-Step Solution

Answer

There are 1,000,000 different ways to choose the six digits required to open the briefcase.

Step 1: Understand the Problem

The problem asks for the number of different 6-digit combinations of numbers that can be set on two locks of a briefcase. Each lock has a 3-digit number, and each digit can range from 0 to 9, with repetition allowed in each lock. Therefore, we need to find the total number of combinations for 2 locks with 3 digits each.

Step 2: Calculate Combinations for One Lock

Each lock can have any of the numbers from 0 to 9 for each of its three digits. Thus, the number of combinations for one lock is given by \(10 \times 10 \times 10 = 10^3\). Evaluate this expression: \[10^3 = 1000\].

Step 3: Calculate Combinations for Two Locks

Since each lock operates independently and has \(1000\) combinations, the total number of combinations for two locks will be the product of each lock's combinations: \[1000 \times 1000\]. Evaluate this expression: \[1000 \times 1000 = 1000000\].

Step 4: Interpret the Result

The total number of different ways to choose the 6 digits required to open both locks is \(1000000\). This accounts for all possible combinations of digits, considering repetitions within each lock.

Key Concepts

Understanding Probability in Combination Locks

Probability gives us a way to quantify the likelihood of different outcomes. In the context of a combination lock, we're interested in understanding the number of possible outcomes, which determines the probability of guessing the correct combination. Here, each lock in the briefcase features digits that range from 0 to 9, allowing repetition. This means each position on the lock can be filled with any of these 10 numbers.

When figuring out the probability of opening a lock by randomly guessing, you consider all possible combinations. For a single 3-digit lock:

• Each digit has 10 possible choices (from 0 to 9).
• The total for one lock is: \(10^3 = 1000\) combinations.

With two locks independently set, the number of total possible outcomes multiplies, giving us a grand total of \(1000000\) combinations. Though the probability of guessing correctly might be low due to these many possibilities, understanding combinations helps in realizing the entire complexity involved.

Exploring Permutations in Multi-digit Combinations

Permutations often come into play when the order of selection matters. However, in combination locks, what we're mainly dealing with is combinations. This is due to the fact that each lock's code is pre-determined by order, which differentiates permutations slightly in this case. Nonetheless, understanding permutations is key to grasp the core concept of ordered choices.

When entering digits, each specific order counts as a unique combination:

• The first digit can be any of the 10 digits,
• The second digit can be any of the 10 digits,
• The third digit can also be any of the 10 digits.

In context of permutations for one lock, \[10 \times 10 \times 10 = 1000\] permutations are possible. This is because order affects the value of the lock combination. With two locks, these permutations multiply, leading to \(1000 \times 1000 = 1000000\) total combinations, due to the independent operation of the locks.

Mathematical Reasoning in Problem Solving

Mathematical reasoning involves logically breaking down a problem into more manageable steps, as seen in this combination lock exercise. By following a step-by-step approach, you simplify complex ideas through straightforward calculations. Here’s how it worked: Firstly, understanding the basic individual lock permutations allows for clear insight into forming overall six-digit combinations. Each lock, being independent, doesn’t influence the other's potential outcomes.

Secondly, breaking down the exercise involved:

• Identifying each lock as a sequence of digits with possible repetitions,
• Using knowledge of how many choices (10 per digit) exist per lock,
• Calculating combinations for one lock, then multiplying by the second lock's outcomes to find the total.

In conclusion, this form of reasoning not only aids in solving mathematical exercises involving combinations and permutations but also ensures clarity in how to logically approach and deduce results. This clarity assists students in tackling similar problems across mathematics.