Problem 37 - Algebra 2 and Trigonometry

Question

Six students will be seated in a row in the classroom. a. How many different ways can they be seated? b. If one student forgot his eyeglasses and must occupy the front seat, how many different seatings are possible?

Step-by-Step Solution

Answer

a. There are 720 different ways. b. There are 120 different ways.

Step 1: Understanding the Problem

We have 6 students who need to be seated in a row. In part (a), we want to find how many different ways they can be seated. In part (b), one student must sit in a specific 'front' seat.

Step 1: Calculate Total Permutations

To find the total number of ways the students can be seated, we calculate the permutations of the 6 students. This is given by the factorial of 6, denoted by \(6!\).

Step 2: Compute 6-Factorial

Calculate \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

Step 4: Result of Step 2

\(6! = 720\). Therefore, there are 720 different ways to seat the 6 students in a row.

Step 3: Determine Permutations with One Student in Fixed Position

In part (b), one student must be in the 'front' seat. We fix one student in that position, which leaves us with 5 students to arrange in the remaining 5 seats.

Step 4: Calculate Permutations for Remaining Seats

Calculate the permutations for the remaining 5 students using \(5!\).

Step 7: Result of Step 4

\(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Thus, there are 120 different seatings possible when one student is fixed in the front seat.

Key Concepts

Factorial

The concept of factorial, often denoted by the symbol \(!\), is fundamental in the fields of permutations and combinatorics. A factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \). For example, \( 6! \) is calculated as \( 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \). This mathematical operation is crucial when determining the number of different ways to arrange a set of items.

• Factorials apply to natural numbers. They're used to calculate permutations and combinations.
• The factorial of zero, \( 0! \), is defined as 1, which is a useful identity in combinatorics.

Understanding factorials helps solve problems where order or arrangement is essential, such as the seating arrangement problem where the full factorial \( 6! \) is used to determine all possible seatings of six students.

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arranging, and combining elements within a set according to certain rules. It's extensively used in problems where the arrangement or selection of objects is of interest.

• Permutations are one of its key areas, dealing with the arrangement of objects where order matters.
• Combinations, another aspect, concern selection where order does not matter.

In our exercise of seating six students in different ways, combinatorics techniques are used to determine all possible permutations—both unrestricted and those with restrictions, such as fixing a student in a specific spot. Hence, combinatorics provides the framework for identifying and calculating various possibilities in problems involving arrangements.

Permutations with restrictions

Permutations involve arranging items in a specific order. In cases with restrictions, certain conditions must be met during arrangement. In the given exercise, we encounter such a scenario in part (b) where one student's seating position is restricted to the front seat.

• Once a position is fixed due to restriction, the number of arrangements for the remaining items is calculated by considering permutations of the remaining items.
• In our example, fixing one person results in 5 positions left to arrange the remaining students, which is computed as the factorial \(5!\).

Applying the concept of permutations with restrictions helps solve many real-world problems where certain conditions apply, allowing for accurate counting and arrangement in various constraints.