Problem 49 - College Algebra with Modeling and Visualization

Question

How many ways can 7 people sit at a round table? (For a way to be different, at least one person must be sitting next to someone different.)

Step-by-Step Solution

Answer

There are 720 ways for 7 people to sit at a round table.

Step 1: Understanding Circular Permutations

When arranging people in a circle, any rotation of the same arrangement is not considered different. Therefore, with 7 people, we consider 1 person fixed and arrange the remaining 6.

Step 2: Rearranging the Remaining People

Since 1 person is fixed, arrange the remaining 6 people. The number of ways to arrange 6 people linearly is given by the factorial of 6, i.e., \( 6! \).

Step 3: Calculating Factorial 6

Calculate \( 6! \), which is done by multiplying 6 by all the positive integers less than 6: \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).

Step 4: Final Computation

Conduct the multiplication to find \( 6! = 720 \). This is the number of ways to arrange the 7 people around the table.

Key Concepts

Factorial Calculation

Understanding the factorial calculation is crucial for solving problems related to permutations and combinations. A factorial, denoted by the symbol "!", is a product of all positive integers up to a certain number. For example, the factorial of 6, written as \(6!\), is calculated by multiplying 6 by every positive integer less than 6:

• \(6 \times 5 = 30\)
• \(30 \times 4 = 120\)
• \(120 \times 3 = 360\)
• \(360 \times 2 = 720\)
• \(720 \times 1 = 720\)

As a result, \(6! = 720\). This value tells us the total number of ways to linearly arrange 6 distinct people or objects. Factorials grow rapidly as numbers increase, making them vital in calculating permutations and combinations.

Permutation Problems

Permutation problems are a central part of combinatorial mathematics, focusing on the arrangement of objects in specific sequences. In permutation problems dealing with linear arrangements, each permutation represents a unique sequence or order of the objects. However, when dealing with circular permutations, the concept differs slightly. When arranging people around a circular table, as in our exercise with 7 people, rotations of the same arrangement are not distinct. This is because if you rotate the seating configuration, it appears the same; hence, you must fix one position to prevent duplication.For example, when one person is fixed at a table, the task reduces to arranging the remaining people linearly. For 7 people, fixing one position means arranging 6 individuals, leading to \(6!\) permutations.

Combinatorial Mathematics

Combinatorial mathematics provides tools for counting, arranging, and selecting objects. It includes concepts like permutations, combinations, and factorials, important for solving complex counting problems.In many combinatorial problems, it's crucial to determine whether repetition is allowed and if the order matters. If both repetition and order are considered, it's a permutation. Otherwise, it's a combination.In circular permutation scenarios, such as with the round table problem, combinatorics helps us understand that we need to account for rotational symmetry. This means counting permutations in a circle rather than a line by fixing one position. Thus, by applying combinatorial principles to such problems, we discern that the number of ways to arrange objects in a circle is generally given by \((n-1)!\). This applies to the 7 people around the round table, resulting in \(6! = 720\) unique seating arrangements.