Problem 44 - Algebra for College Students

Question

April, Bill, Carl, and Denise are to be seated at random in a row of four chairs. What is the probability that April and Bill will occupy the end seats? \(\frac{1}{6}\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{6} \).

Step 1: Determine Total Arrangements

First, calculate the total number of ways to arrange April, Bill, Carl, and Denise in four chairs. This is a simple permutation problem where you arrange four individuals. The total number of permutations is given by: \[ 4! = 24 \] This means there are 24 different ways to arrange four people in a row.

Step 2: Identify Specific Condition

Next, we want to find how many ways April and Bill can sit in the end seats. There are two end seats and two people (April and Bill) to choose from, so we can swap them: \[ 2! = 2 \] This means we have 2 arrangements for who sits in the end seats (April could sit left and Bill right, or Bill left and April right).

Step 3: Arrange Remaining People

Once April and Bill are sitting in the end seats, Carl and Denise will occupy the two middle seats. The number of arrangements for Carl and Denise is: \[ 2! = 2 \] So, Carl and Denise can be seated in the middle seats in 2 different ways.

Step 4: Calculate Favorable Arrangements

Calculate the total number of favorable arrangements by multiplying the choices for the end seats by the choices for the middle seats: \[ 2 \times 2 = 4 \] So, there are 4 different ways where April and Bill sit in the end seats.

Step 5: Compute Probability

Finally, calculate the probability by dividing the total number of favorable arrangements by the total number of possible arrangements:\[ \frac{4}{24} = \frac{1}{6} \]Therefore, the probability that April and Bill occupy the end seats is \( \frac{1}{6} \).

Key Concepts

Probability Concepts

Understanding probability is important in determining the likelihood of an event occurring. Probability is the measure of how likely an event is to happen, often expressed as a fraction or a percentage. It ranges from 0 (impossible event) to 1 (certain event). For discrete events, the probability can be calculated using the formula: \[ P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}} \]In the exercise about April, Bill, Carl, and Denise, it asks us to determine the probability that April and Bill occupy the end seats. Here, the favorable outcomes are the specific ways April and Bill can sit in the desired positions, whereas the total possible outcomes are every possible arrangement of the four friends in the seats.

• Count the favorable outcomes.
• Calculate the total possible outcomes.
• Use the probability formula to find the probability.

By using these steps, we determine the likelihood of April and Bill sitting in the end seats as \( \frac{1}{6} \). This exercise thus explores a basic probability concept through arrangement scenarios.

Arrangement Problems

Arrangement problems, like the one with April, Bill, Carl, and Denise, focus on how different objects, people, or elements can be ordered or organized. When addressing such problems, it's important to consider whether the order in which items are arranged matters. Situations where order matters involve permutations, whereas combinations are used when order does not matter. Here, we are dealing with a permutation problem because we care about specific seating arrangements, particularly who sits at the ends and the middle. The original exercise is solved by first calculating all possible ways to arrange the four individuals in chairs, which is a typical step in solving arrangement problems. The next step involves focusing on a subset of these arrangements - specifically those where April and Bill are in the end positions. Using simple calculations and logical steps, we can narrow down specific configurations, like swapping April and Bill and determining how Carl and Denise can sit. With arrangement problems:

• Determine if order matters (use permutations if yes, combinations if no).
• Identify each step needed to narrow down the configurations based on given conditions.
• Count each possible arrangement by evaluating specific conditions or restrictions.

This process helps us achieve the desired arrangement outcomes easily.

Basic Permutation Calculations

Permutations are fundamental in understanding arrangements where order is paramount. A permutation calculates the total number of ways to arrange a set number of items. The mathematical representation for permutations of \( n \) distinct items is given by the factorial of \( n \), represented as \( n! \).In the problem, determining permutations for four people is straightforward; use \( 4! \). This corresponds to:\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]This shows that there are 24 different possible ways to arrange the four friends in a row.When certain constraints are applied, like specific seating positions for April and Bill, the exercise illustrates the application of permutations on a smaller subset. Applying constraints reduces the number of permutations to a manageable number, simplifying complex problems. For example, fixing April and Bill’s positions immediately reduces possibilities to permutations of Carl and Denise for the remaining seats:

• Compute general permutations without restrictions (use factorial calculation).
• Apply the problem-specific restrictions to narrow the permutations down.
• With fixed positions, calculate new arrangements for remaining items.

This method helps in strategically solving arrangement issues by breaking the problem down step by step.