Question

A passenger train consists of two baggage cars, four day coaches, and three parlor cars. In how many ways can the train be made up if the two baggage cars must be in the front and the three parlor cars must be in the rear? Assume that the baggage cars can be told apart, that the day coaches can be told apart, and that the parlor cars can be told apart.

Short answer

There are 288 ways to arrange the train.

Step-by-step solution

Identify Train Segment Arrangements

First, we need to arrange the sections of the train. The train has three major sections: the two baggage cars at the front, followed by the four day coaches, and finally the three parlor cars at the rear. The sections themselves do not change positions.

Arrange the Baggage Cars

The two baggage cars need to be arranged. Since they are distinguishable, the number of ways to arrange them is given by the factorial of the number of baggage cars: \[2! = 2\] ways.

Arrange the Day Coaches

The day coaches are in the middle section of the train. Since all four day coaches can be told apart, the number of ways to arrange them is: \[4! = 24\] ways.

Arrange the Parlor Cars

Finally, the three parlor cars at the rear must be arranged. They too are distinguishable, so the number of ways to arrange them is:\[3! = 6\] ways.

Calculate Total Arrangements

Multiply the number of ways to arrange each section to get the total number of ways to arrange the entire train.\[2! \times 4! \times 3! = 2 \times 24 \times 6 = 288\] ways.

Key concepts

Factorials

Factorials are a key concept in combinatorics and mathematics, often denoted by an exclamation mark (!). The factorial of a non-negative integer, say \( n \), is the product of all positive integers less than or equal to \( n \). For example, if you want to find the factorial of 5 (denoted as \( 5! \)), you would calculate \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Factorials are used to calculate the number of ways to arrange a set of objects. Because each object needs to be placed in a unique position, and each arrangement involves placing subsequent objects in the remaining spots, factorials simplify this process.

• For a single item (1!), there's only one way to do this.
• For two items (2!), you have two possible swaps: first and second or second and first.
• Continue this up to n items, and you see how factorials encompass all possible orders.

Factorials grow rapidly with higher numbers, making them essential for counting arrangements in combinatorial problems like arranging train cars.

Permutations

Permutations refer to the different ways of arranging a set of objects in sequential order. Unlike combinations, where order does not matter, order is paramount in permutations.

To find the permutations of a group of items, you will frequently use factorials. For instance, if you want to know how many ways you can arrange 4 distinguishable day coaches in a train, you would use a permutation formula, which is \( n! \) where \( n \) is the number of items. Here, it's \( 4! = 24 \) permutations.

• Each distinct order is a different permutation.
• If all items are distinct, the number of permutations is simply the factorial of the number of items.
• Permutations are key when order impacts the result, such as arranging unique carriages in a train.

Permutations are foundational in solving problems where sequential order plays a critical role, making conceptual understanding vital in discrete mathematics.

Discrete Mathematics

Discrete Mathematics is a branch of mathematics dealing with discrete elements that use distinct values. This includes studies like graph theory, counting principles, and combinatorics.

Within combinatorics, discrete mathematics explores different methods and formulas to solve problems about arranging and selecting objects, known as permutations and combinations. The problems are usually non-continuous and focus on integers and finite sets. For instance, determining how train cars can be arranged is a discrete problem because it involves a finite number of distinguishable cars.

• Deals with countable, separate objects.
• Includes topics like logic, set theory, and combinatorics.
• Aids in understanding structure and symmetry within baked-in rules.

Understanding discrete mathematics helps break down complex arrangements into understandable steps, as seen in arranging train cars while maintaining specified orders for each section.

Train Arrangement Problem

The Train Arrangement Problem is a real-world application of combinatorial principles. In this problem, you must arrange different types of train cars, such as baggage, day coaches, and parlor cars, adhering to specific placement rules.

In the exercise, the baggage cars must be up front, the day coaches in the middle, and the parlor cars at the rear. All are distinguishable, allowing for multiple arrangements. Solving it involves:

• Arranging each type of car using permutations.
• Applying factorials to determine the unique sequences for each section.
• Multiplying the factorial results from each segment to find the total arrangement possibilities.

This kind of problem exemplifies how permutations and discrete mathematics simplify complex organizational tasks, providing clarity in scenarios like managing a passenger train with different arrangements.