Problem 31 - Algebra and Trigonometry

Question

19–32 These problems involve permutations. Seating Arrangements In how many ways can five students be seated in a row of five chairs if Jack insists on sitting in the first chair?

Step-by-Step Solution

Answer

There are 24 ways to seat the students with Jack in the first chair.

Step 1: Identify the Fixed Element

Jack insists on sitting in the first chair, which means we have one student whose position is fixed. Therefore, we only need to arrange the remaining four students.

Step 2: Determine the Arrangement of the Remaining Students

Since Jack's position is fixed, we need to arrange the remaining four students in the four remaining chairs. We use the permutation formula for the number of ways to arrange n distinct items: \( n! \).

Step 3: Apply the Permutation Formula

Calculate the number of arrangements for the four students using the factorial of 4 (since n = 4): \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Step 4: Consider All Possible Arrangements

Since Jack's position is fixed, the only possible arrangements are the 24 arrangements of the remaining four students.

Key Concepts

Factorial

The concept of a factorial is crucial in many mathematical fields, especially in permutations and combinations. A factorial of a non-negative integer, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). It is a way to determine how many ways you can arrange a set quantity of items.

For instance, the factorial of 4 is calculated as follows:

• Begin with the number 4 and multiply it by every smaller integer down to 1.
• So, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Knowing how to calculate factorials helps determine the total possible arrangements of a set, which is particularly useful in permutations. Factorials grow very quickly with larger numbers, making them essential for calculating vast quantities of arrangements efficiently.

Combinatorics

Combinatorics is the branch of mathematics that studies, in essence, counting. It is the science of counting arrangements and helps us explore the possible configurations of a set of items. In this context, it aids in understanding permutations, combinations, and other forms of arrangements.

One of the foundational ideas in combinatorics is determining how to count elements easily and efficiently. When dealing with permutations, combinatorics allows us to find the number of possible sequences without having to physically arrange the items in every possible way. Instead, using formulas and techniques, like factorial calculations, helps simplify this problem, ensuring that students can focus on understanding the underlying principles and concepts.

Arrangement Problems

Arrangement problems are a subset of combinatorial problems where the goal is to determine how items can be ordered or sequenced. These often appear in real-world scenarios, such as seating plans, tournament schedules, or even solving puzzles.

In these problems, constraints such as fixed positions or preferred orders can add layers of complexity. For instance, in our example where Jack must sit in the first chair, the arrangement problem simplifies since only the remaining students are considered for different arrangements.

• First, identify any fixed elements or constraints, such as Jack's need to sit in a specific chair.
• Next, use permutations (like the factorial method) to determine the number of arrangements for the remaining items, in this case, students.

Understanding arrangement problems effectively requires a combination of recognizing set constraints and applying combinatorial techniques to explore all feasible options.