Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Use the fundamental principle of counting or permutations to solve each problem. In an experiment on social interaction, 6 people will sit in 6 seats in a row. In how many ways can this be done?

Step-by-Step Solution

Answer

720 ways.

Step 1: Understanding the Problem

We need to determine the number of ways to arrange 6 people in 6 seats. This is a problem of arranging objects in a specific order.

Step 2: Apply the Concept of Permutations

When arranging n distinct objects in a sequence, we use permutations. For n objects, the number of permutations is calculated as \( n! \) (n factorial). Here, n is 6, so 6 people can be arranged in \( 6! \) different ways.

Step 3: Calculate 6 Factorial

Calculate \( 6! \), which is the product of all positive integers from 6 down to 1: \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).

Step 4: Compute the Product

Perform the multiplication: \( 6 \times 5 = 30 \), \( 30 \times 4 = 120 \), \( 120 \times 3 = 360 \), \( 360 \times 2 = 720 \), \( 720 \times 1 = 720 \).

Step 5: Conclusion

There are 720 different ways to arrange 6 people in 6 seats.

Key Concepts

Permutations

Permutations represent the different ways we can arrange a set of items in order, where the order itself matters. Imagine you have a set of books, and you want to place them on a shelf. The order in which you place them changes the permutation.

• If you have 3 books, you can arrange them in the order ABC, ACB, BAC, BCA, CAB, or CBA, resulting in 6 permutations in total.
• An easy way to remember this is that permutations are all about different possible sequences.

This concept is used extensively in counting problems where distinguishing between different sequences is essential.

Why Order Matters

When it comes to permutations, having a different order means having a different outcome. If you change the order of people sitting in a row, it forms a new permutation. This is why permutations are distinct from combinations, where the order does not matter.

Factorial

The factorial of a number, denoted by the exclamation mark symbol (!), is the product of all positive integers up to that number.

• For example, the factorial of 5, written as 5!, is calculated through the multiplication of all whole numbers from 1 to 5: 5! = 5 × 4 × 3 × 2 × 1 = 120.
• Remember, the factorial of zero (0!) is defined to be 1.

Factorials are crucial for calculating permutations because they provide the total number of ways to arrange a set number of items in different orders.

How Factorials Work in Permutations

When calculating permutations, the formula used is: \[P(n) = n! \]For arranging 6 people in 6 seats, you calculate 6! because each person can take any seat, leaving a specific number of seats for the rest. Factorials make it easier to handle these computations by giving us a straightforward multiplication process.

Arrangements

Arrangements refer to the specific ordering or sequences of items, often within a specific framework, such as seating people in predefined seats. The concept of arrangements is directly linked to permutations, as both describe an orderly sequence.

• In scenarios like seating people in specific chairs, arrangements allow us to explore the numerous sequences available.
• Given 6 seats and 6 people, different arrangements mean each seat can be occupied by different individuals in varying order.

Counting arrangements involves permutations because the order of seating changes the scenario.

Exploring Different Arrangements

To visualize this, consider musical notes on a piano where arrangement defines different melodies. Similarly, arranging people in a row leads to unique sequences for each combination.