Problem 15 - Combinatorics Through Guided Discovery

Question

A roller coaster car has \(n\) rows of seats, each of which has room for two people. If \(n\) men and \(n\) women get into the car with a man and a woman in each row, in how many ways may they choose their seats?

Step-by-Step Solution

Answer

The number of ways to choose the seats is n!.

Step 1: Understand the Problem

There are n rows of seats, and each row can seat two people. We need to find the number of ways to seat n men and n women such that each row has one man and one woman.

Step 2: Seat Men in the Rows

First, consider the n men. These men can sit in any of the n rows. Therefore, there is only one way to place each man in a specific row.

Step 3: Seat Women in the Remaining Seats

For each arrangement of the men in their seats, we need to place the n women in the remaining seats such that each row contains one man and one woman. Each woman can be seated in any of the remaining seats without restriction.

Step 4: Calculate the Number of Ways to Seat the Women

Since we need to choose which woman will sit with each man, we can think of arranging the women in the remaining n seats as a permutation problem. The number of permutations of n women is given by n factorial (!).

Step 5: Conclusion

The total number of ways to seat n men and n women such that each row contains one man and one woman is given by n!. Thus, the answer is n!.

Key Concepts

permutations

Permutations are a fundamental concept in combinatorics. They refer to the different ways in which a set of objects can be arranged. For instance, if you have a set of three letters: A, B, and C, permutations of these letters would include: ABC, ACB, BAC, BCA, CAB, and CBA.

To calculate the number of permutations of a given set, we use the concept of factorials. A permutation of a set with n elements is denoted as n!, which stands for 'n factorial'. Factorials will be discussed in detail in the next section.

In our roller coaster seating problem, we are using permutations to figure out how to arrange the women in the seats since each row already has a seat reserved for a man. This arrangement results in n! different ways to seat the women.

factorials in combinatorics

A factorial, denoted by the symbol (!), is the product of all positive integers up to a certain number. For example, 5! is calculated as:
5! = 5 × 4 × 3 × 2 × 1 = 120

Factorials are very important in combinatorics because they provide a way to calculate the number of permutations and combinations. For an integer n, the factorial n! represents the total number of ways to arrange n distinct items.

In our problem, we are using this concept to determine how many different ways there are to seat n women in the remaining seats after seating the men. The total number of ways to arrange the women is given by n!, signifying all possible permutations.

This simplifies the problem greatly, as we only need to calculate one value: n!.

combinatorial problem solving

Combinatorial problem solving is a method used to count and analyze possible arrangements of objects within certain constraints. It involves using concepts like permutations and combinations to find the number of ways a task can be completed.

In the roller coaster seating problem, we start by identifying the constraints: Each row of seats must contain one man and one woman.

Step-by-step, we solve the problem as follows:
* First, we seat all the n men. This part is straightforward because there’s exactly one spot for each man.
* Second, we focus on seating the women in the remaining seats. Here we use permutations, as each woman can sit next to any of the seated men.

By framing the seating of women as a permutation problem, we see that the number of ways to arrange the women is precisely n!.

Combinatorial problem solving thus simplifies our approach and allows us to systematically count and understand possible arrangements.