Problem 19 - Algebra for College Students

Question

Aimée, Brenda, Chuck, Dave, and Eli are randomly seated in a row of five seats. Find the probability that Aimée and Chuck are not seated side by side. \(\frac{3}{5}\)

Step-by-Step Solution

Answer

The probability that Aimée and Chuck are not seated side by side is \(\frac{3}{5}\).

Step 1: Calculate Total Number of Arrangements

First, calculate the total number of ways to arrange five people in five seats. This is a permutation problem, which can be calculated using the factorial of the number of people, i.e., \(5!\). So, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

Step 2: Calculate Arrangements with Aimée and Chuck Together

To find the number of arrangements where Aimée and Chuck are seated side by side, treat them as a single unit or block. This block can be arranged in \(4!\) different ways due to the other three people (Brenda, Dave, Eli), resulting in \(24\) ways. Within their block, Aimée and Chuck can be arranged in \(2!\) ways, giving \(2! = 2\) ways. Therefore, there are \(4! \times 2 = 48\) arrangements where Aimée and Chuck sit together.

Step 3: Calculate Arrangements with Aimée and Chuck Not Together

Subtract the number of arrangements where Aimée and Chuck are together from the total number of arrangements to find how many arrangements have Aimée and Chuck not seated next to each other. So, it's \(120 - 48 = 72\) ways.

Step 4: Find the Probability

The probability that Aimée and Chuck are not seated side by side is the number of ways they are not together divided by the total number of arrangements. Therefore, the probability is \(\frac{72}{120}\). Simplify this fraction by dividing the numerator and denominator by their greatest common divisor, which is 24, resulting in \(\frac{3}{5}\).

Key Concepts

Factorial Calculation

In probability theory and combinatorics, factorial calculations are often the foundational step for determining the number of possible arrangements or outcomes in a given situation. The factorial of a non-negative integer, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, for \( n = 5 \), the factorial calculation is \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Factorials are crucial in arranging objects where the order matters. It's like figuring out how to place books on a shelf or people in a photo lineup. Each unique sequence or arrangement represents a different permutation, which we'll explore further in the next section.

Permutations

Permutations deal with the arrangement of objects where the order is important. In the context of our exercise, it involves arranging people, Aimée, Brenda, Chuck, Dave, and Eli, in a row of seats. If we treat each person as a unique item, then arranging \( n \) distinct items is a permutation problem calculated using \( n! \). In our case, this is \( 5! \).

However, sometimes permutations are restricted, like when specific people must sit together. When Aimée and Chuck are treated as a unit, we arrange them with the other three people, resulting in \( 4! \) permutations, since we're dealing with four groups now. But don't forget, within their unit, Aimée and Chuck can switch places; thus, there are \( 2! \) ways to arrange them within their block. These adjustments are what make permutation calculations versatile.

Combinatorics

Combinatorics is a field of mathematics focused on counting, arranging, and analyzing combinations and permutations of sets. It applies crucial principles to arrange elements under specified constraints, like the requirement that Aimée and Chuck should or shouldn't sit together.

This field helps solve various problems in probabilistic scenarios by simplifying complex arrangements into simpler sub-problems. By breaking it down, you can calculate the total number of arrangements possible (using factorials) and further manipulate these calculations to find subsets that meet certain criteria. Combinatorics allows us to precisely determine quantities like the arrangements where Aimée and Chuck do not side each other, as solved through subtraction in our exercise.

Arrangement Probability

In probability theory, arrangement probability often refers to the likelihood of one specific arrangement occurring among all possible arrangements. For problems involving specific constraints, like seating policies, calculating probabilities involves determining how many favorable arrangements exist compared to the total possible arrangements.

In our exercise, we first identified all possible arrangements: \( 5! \) for five people. We then found arrangements where Aimée and Chuck sit together and subtracted them from the total. Finally, we calculated the probability that they are not seated together, which turns out to be \( \frac{3}{5} \). This involves taking the favorable outcomes for our condition (where they are not together) and dividing by the total possible outcomes. Such calculations are pivotal in statistics and many real-world applications like seating, scheduling, and resource allocation.