Problem 25 - Algebra 2

Question

Determine whether each situation involves a permutation or a combination. Then find the number of possibilities. selecting two of eight employees to attend a business seminar

Step-by-Step Solution

Answer

It is a combination, with 28 possibilities.

Step 1: Understand the Situation

In this exercise, we must determine if the situation is a permutation or a combination. We need to select two employees out of eight to attend a seminar. The order of selection does not matter, so this situation is a combination, not a permutation.

Step 2: Use the Combination Formula

The formula for combinations is \( C(n, k) = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items to choose from and \( k \) is the number of items to choose. Here, \( n = 8 \) and \( k = 2 \).

Step 3: Compute the Factorials

Calculate the factorials needed for the combination formula: \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \), \( 2! = 2 \times 1 \), and \( (8-2)! = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).

Step 4: Plug Values into the Formula

Substitute the factorial values into the combination formula: \( C(8, 2) = \frac{8!}{2!6!} = \frac{8 \times 7 \times 6!}{2! \times 6!} \). The \( 6! \) terms cancel each other out.

Step 5: Simplify the Expression

After cancellation, the expression simplifies to \( \frac{8 \times 7}{2 \times 1} = \frac{56}{2} \), which equals 28.

Key Concepts

Permutations

A permutation refers to arranging items in a specific order. When a situation calls for a permutation, the sequence of selection matters. For instance, if you are arranging books on a shelf, the order in which the books are placed defines the different permutations. It's about finding out how many ways you can arrange a set of items where the order alters the outcome. In permutations, each arrangement is unique because the arrangement order counts:- The number of permutations of selecting and arranging two people out of a group of eight differs from simply selecting them without order.- If order was crucial in our original exercise, we'd use permutations, not combinations.The formula for permutations is given by: \[ P(n, k) = \frac{n!}{(n-k)!} \] where \( n \) is the total number of items and \( k \) is the number of items to arrange. Remember, permutations are used when the sequence is important. If not, we lean on combinations, as shown in the exercise.

Factorials

Factorials are a fundamental concept in permutations and combinations. The factorial of a positive integer \( n \) is the product of all positive integers less than or equal to \( n \). It is symbolized as \( n! \). Let's break it down:- For \( 8! \), calculate as: \( 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).- Factorials grow rapidly in size, meaning they are large numbers even for modest \( n \) values.Factorials play a crucial role in determining combinations and permutations:- In combinations, they help determine how many different groups can be formed.- In permutations, they tell us how many different sequences can be made.Using factorials enables the use of mathematical formulas effectively, ensuring calculations for combinations and permutations are feasible.

Combination Formula

The combination formula calculates the number of ways to select \( k \) items from \( n \) items without regard to the order of selection. This is important in scenarios where the sequence does not matter, such as picking team members or in our problem of selecting seminar participants.The combination formula is expressed as: \[ C(n, k) = \frac{n!}{k! (n-k)!} \] Here's how it works in practical terms:- Use \( n! \) to calculate the total ways of arranging \( n \) items.- Divide by \( k! \) to remove the arrangements of the \( k \) items themselves.- Divide by \( (n-k)! \) to take out the arrangements of the unselected items.This formula directly applied allows one to calculate combinations without error and is fundamental in situations where order is not priority.

Mathematics Education

Understanding concepts like permutations and combinations is vital in mathematics education. These foundational concepts are not just theoretical but have practical applications in probability, computer science, and various decision-making processes. Incorporating real-world examples, like the exercise provided, enhances learning by linking mathematical concepts to everyday situations: - Engages students more effectively by showing use-case scenarios. - Helps in developing problem-solving skills applicable beyond academic settings. Effective mathematics education involves explaining tough concepts in easily understandable forms, often using visual aids and step-by-step instructions: - Encourages student participation and active learning. - Builds confidence as students solve complex problems with straightforward strategies. By breaking down each element, educators can ensure students fully grasp permutations, combinations, and factorials, preparing them for advanced studies.