Problem 29 - Algebra 2 and Trigonometry

Question

In how many ways can 3 different job openings be filled if there are 8 applicants?

Step-by-Step Solution

Answer

There are 336 ways to fill the 3 job openings with 8 applicants.

Step 1: Understand the Problem

We need to determine how many different ways three distinct job openings can be filled from a pool of 8 candidates. The order in which the jobs are filled matters (since the jobs are different), so this is a permutation problem.

Step 2: Apply the Permutation Formula

The number of permutations of choosing 3 positions from 8 applicants can be determined using the permutation formula for choosing and arranging r items out of n: \[ P(n, r) = \frac{n!}{(n-r)!} \]Here, \( n = 8 \) and \( r = 3 \).

Step 3: Calculate the Factorials

Calculate the factorial terms needed for the formula:- First, calculate the factorial of n: \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)- Then, calculate the factorial of \((n-r)\): \( (8-3)! = 5! = 5 \times 4 \times 3 \times 2 \times 1 \)

Step 4: Substitute into the Formula

Substitute these values into the permutation formula:\[ P(8, 3) = \frac{8!}{5!} = \frac{8 \times 7 \times 6 \times 5!}{5!} \]Simplifying, the \( 5! \) in the numerator and denominator cancels out.\[ P(8, 3) = 8 \times 7 \times 6 \]

Step 5: Perform the Multiplication

Multiply the numbers from the simplified expression:\[ 8 \times 7 \times 6 = 336 \]

Key Concepts

Factorials

Factorials are mathematical expressions used to simplify calculations, particularly in permutations and combinations. A factorial of a positive integer \( n \), represented by \( n! \), is the product of all positive integers from 1 to \( n \). For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factorials grow rapidly as the number \( n \) increases. They are used extensively in permutations to calculate the total number of ways to arrange a certain number of items. In our original exercise, we calculated \( 8! \) to help find how many different ways we can arrange 8 applicants into 3 job openings.
Quick tips for working with factorials:

• Remember that \( 0! = 1 \) by definition, which is an important edge case.
• Factorials are only defined for non-negative integers.
• If you spot a factorial in both the numerator and denominator, see if they cancel out to simplify calculations.

Permutation Formula

The permutation formula allows us to calculate the number of ways to arrange \( r \) items chosen from \( n \) total items, where the order matters. This formula is especially useful when assigning different job roles or selecting winners for prizes, like in our job applicant scenario.
The permutation formula is given by:\[ P(n, r) = \frac{n!}{(n-r)!} \]Where:

• \( n \) is the total number of items to choose from (8 applicants).
• \( r \) is the number of items to arrange (3 job openings).

To apply the formula, we substitute the values to get \( P(8, 3) = \frac{8!}{5!} \). This simplifies to \( 8 \times 7 \times 6 = 336 \) by canceling the common \( 5! \) in numerator and denominator.
Remember that permutation formulas are suitable only when the arrangement order is important. If the arrangement doesn't matter, combinations should be used instead.

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arrangement, and combination of objects. It plays a crucial role in probability, algebra, and computer science. In essence, combinatorics helps us efficiently count different configurations based on given criteria.
There are two major types of combinatorial arrangements:

• Permutations: Used when order matters. As seen with our job openings, each different order of placing applicants into positions counts as a unique possibility.
• Combinations: Used when order does not matter. If we were only interested in the group of selected applicants without concern for position, combinations would be used.

Understanding these concepts is vital for tackling real-world problems involving organization, quality control processes, and even games and puzzles. Mastering combinatorics can enhance analytical thinking and problem-solving skills.