Problem 25 - Algebra and Trigonometry

Question

These problems involve permutations. Seating Arrangements In how many different ways can six of ten people be seated in a row of six chairs?

Step-by-Step Solution

Answer

There are 151200 ways to seat six of ten people in a row.

Step 1: Understanding Permutations

Permutations involve arranging a certain number of items from a larger set where the order of arrangement matters. In this problem, we're arranging 6 out of 10 people.

Step 2: Setting Up the Formula

To find the number of ways to arrange 6 people from a set of 10, we use the permutation formula: \[ P(n, r) = \frac{n!}{(n - r)!} \]where \( n \) is the total number of people (10 in this case) and \( r \) is the number of people to arrange (6 in this case).

Step 3: Calculate Factorials

Calculate the factorials needed: - \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)- \(4! = 4 \times 3 \times 2 \times 1\)

Step 4: Compute the Permutation

Plug the factorials into the permutation formula: \[ P(10, 6) = \frac{10!}{(10 - 6)!} = \frac{10!}{4!} \]This simplifies to: \[ P(10, 6) = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4!}{4!} \]Since the \(4!\) cancels out, we are left with: \[ 10 \times 9 \times 8 \times 7 \times 6 \times 5 \]

Step 5: Perform the Multiplication

Calculate the result of the multiplication: \[ 10 \times 9 \times 8 \times 7 \times 6 \times 5 = 151200 \]

Step 6: Conclude the Calculation

Thus, the number of different ways to seat six of the ten people in a row of six chairs is 151200.

Key Concepts

Seating Arrangements

When it comes to seating arrangements, the order in which individuals are seated matters a lot. Imagine having ten friends and wanting to seat six of them in a row. The specific order in which these six friends are arranged leads to different seating combinations. For example, switching the first and second seat changes the entire arrangement. This is why we use permutations, which focus on ordered arrangements. Each unique order is called a permutation.

Permutation Formula

The permutation formula helps us calculate the number of ways to arrange a subset of items from a larger set. It is expressed as \[ P(n, r) = \frac{n!}{(n-r)!} \]where:

• \( n \) is the total number of items to choose from.
• \( r \) is the number of items to arrange.

In our given exercise, we have 10 people, and we want to arrange 6 of them. By using the permutation formula, we find the number of possible seating arrangements.

Factorials

Factorials are essential in calculating permutations. A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a certain number. For example,

• \(5! = 5 \times 4 \times 3 \times 2 \times 1\)
• \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)

In permutations, we calculate the factorial of the total number of items (\( n! \)) and divide it by the factorial of the difference (\((n-r)!\)) to find the number of ordered arrangements. Factorials grow rapidly, which is why they are incredibly useful in combinatorics.

Combinatorics

Combinatorics is the branch of mathematics dealing with counting, arrangement, and combination of elements in sets. It answers questions like "How many ways can these people be seated?". By understanding combinatorics, we learn to solve problems involving ordering and selecting items. This field uses concepts like permutations and combinations, allowing us to tackle various arrangement puzzles efficiently. Whether it's seating friends or organizing events, combinatorics provides the tools needed to find the best arrangement.