Problem 6 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Four commuter trains and three express buses depart from city A to city B in the morning, and three commuter trains and three express buses operate on the return trip in the evening. In how many ways can a commuter from city A to city B complete a daily round trip via bus and/or train?

Step-by-Step Solution

Answer

There are 42 ways in which a commuter from city A to city B can complete a daily round trip via bus and/or train.

Step 1: Morning Commute Options

First, let's determine the number of ways the commuter can travel from city A to city B in the morning. As mentioned earlier, four commuter trains and three express buses are available for the morning commute. Since the commuter can choose any one of these vehicles, we have a total of 4 + 3 = 7 morning commute options.

Step 2: Evening Commute Options

Next, we will determine the number of ways the commuter can travel back from city B to city A in the evening. There are three commuter trains and three express buses available for the evening commute. Thus, the total number of evening commute options is 3 + 3 = 6.

Step 3: Finding the Total Number of Daily Round Trip Options

Now that we've determined the number of morning and evening commute options, we can use the concept of combinations to find the total number of ways to complete a daily round trip. Since the commuter can choose any morning commute option and any evening commute option independently, we simply need to multiply the number of morning options by the number of evening options. So, we have 7 morning options × 6 evening options = 42 possible daily round trip options. Therefore, there are 42 ways in which a commuter from city A to city B can complete a daily round trip via bus and/or train.

Key Concepts

Counting Principle

The counting principle, also known as the fundamental counting principle, allows us to calculate the total number of outcomes when there are several stages or events, and the outcome of each stage does not affect the outcomes of the others. This principle states that if one event can occur in 'm' ways, and another independent event can occur in 'n' ways, then the total number of ways both events can occur is the product of 'm' and 'n'.

For example, in the textbook exercise regarding daily round trips from city A to city B, we had two independent events: traveling to city B in the morning, with 7 options, and returning to city A in the evening with 6 options. Applying the counting principle, the total number of round trip options is the product of the options for each leg: 7 options in the morning times 6 options in the evening equal 42 possible daily round trip options. This concept is both fundamental and immensely practical in combinatorics.

Combinations

Combinations are a core concept in combinatorics that deal with selecting items from a group where order does not matter. When we use combinations, we're often answering questions like 'How many ways can I choose a committee of 4 people from a group of 10?' To calculate combinations, we would use the formula: \[C(n, k) = \frac{n!}{k!(n - k)!}\], where 'n' represents the total number of items to choose from, 'k' is the number of items to be chosen, and '!' denotes the factorial.

However, in our example about the daily round trip, the order does matter since the morning and evening trips are distinct events. Therefore, the concept of combinations isn't directly applied to solving this problem. Yet, understanding combinations is vital when confronting problems where the sequence of selection or order of occurrence isn't important.

Permutations

Permutations, unlike combinations, are used when the order does matter. They can be calculated for a set of distinct items by using the formula: \[P(n) = n!\] for all the items or \[P(n, k) = \frac{n!}{(n - k)!}\] when selecting 'k' items from 'n' available options. The exclamation point represents the factorial of a number, meaning the product of all positive integers up to that number.

One of the common misconceptions in combinatorics is confusing permutations and combinations. Though our round trip problem is not solved using permutations, it serves to highlight that if we needed to consider the order of the buses or trains taken (for example, if each vehicle was distinctly labeled and riders cared about which specific buses or trains they took), we would explore permutation concepts to solve the problem.

Applied Mathematics

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Combinatorics falls under applied mathematics because it often translates real-life scenarios into mathematical terms to find logical solutions. It's the reasoning used in optimization, decision-making, and strategic planning.

Our textbook problem is an example of how applied mathematics functionalizes abstract concepts of combinatorics to answer practical questions, such as finding the number of daily transportation options. The simplicity of using the counting principle in this scenario illustrates the power of applied mathematics to make daily life decisions more manageable and fact-based.