Problem 50 - Algebra for College Students

Question

Randomly select one of the 120 permutations of the letters \(a, b, c, d\), and \(e\). Find the probability that in the chosen permutation, the letter \(a\) precedes the \(b\) (the \(a\) is to the left of the \(b\) ). \(\frac{1}{2}\)

Step-by-Step Solution

Answer

The probability is \(\frac{1}{2}\).

Step 1: Understand the Problem

We are given a set of 5 letters: \(a, b, c, d, e\). We need to determine the probability that, in a randomly chosen permutation of these letters, the letter \(a\) appears before \(b\).

Step 2: Calculate the Total Number of Permutations

Since there are 5 distinct letters, the total number of permutations is \(5! = 120\).

Step 3: Identify Favorable Outcomes

Any permutation where \(a\) is to the left of \(b\) is favorable. For each pair of \(a\) and \(b\), either \(a\) is before \(b\), or \(b\) is before \(a\). They are equally likely, so there are as many permutations with \(a\) before \(b\) as there are with \(b\) before \(a\).

Step 4: Divide the Permutations

Every permutation of the 5 letters can be split into two equal groups: one where \(a\) precedes \(b\) and another where \(b\) precedes \(a\). Thus, each group contains \(\frac{120}{2} = 60\) permutations.

Step 5: Calculate Probability

The probability that \(a\) comes before \(b\) is the number of favorable permutations divided by the total permutations, which is \(\frac{60}{120} = \frac{1}{2}\).

Key Concepts

Permutations

Permutations are a fundamental concept in mathematics that revolve around the concept of arranging elements from a set. More specifically, a permutation refers to any possible order or arrangement of a set of items. For instance, when we have a set of five distinct letters \(a, b, c, d, e\), a permutation would be any specific order in which these letters can appear.
To calculate the number of possible permutations of a set, we use the factorial function, denoted by \(!\). For our set of 5 letters, the total number of permutations is calculated by \(5!\). This is equivalent to multiplying all positive integers up to 5, resulting in \(5 \times 4 \times 3 \times 2 \times 1 = 120\) total permutations.
Permutation calculations help us understand how different arrangements, like how "abcde" can change to "edcba" or "baced", are formed. Such understanding is crucial, especially when dealing with problems where order matters.

Combinatorics

Combinatorics is an area of mathematics dealing with the counting, arrangement, and combination of objects. While permutations focus on ordered arrangements, combinatorics allows us to explore a broader field that includes different methods of grouping and choosing items.
In our permutation problem, combinatorics is essential in understanding how many times a specific condition, such as letter \(a\) preceding \(b\), can occur. We first identify that every sequence of these letters, which is one of their permutations, is equally likely.
When considering our set of letters, we have two competing scenarios for just \(a\) and \(b\):

• \(a\) is before \(b\), or
• \(b\) is before \(a\)

Thanks to combinatorics, we deduce that both events have equal chances, meaning half the permutations will have \(a\) before \(b\). This approach simplifies the calculation and illustrates the power of combinatorial reasoning.

Probability Theory

Probability theory studies the likelihood of events occurring within a given framework. In this exercise, it helps us estimate the chance of a particular arrangement of letters happening by random selection.
Here, the probability of \(a\) preceding \(b\) out of the permutations involves calculating how many favorable outcomes (permutations where \(a\) is before \(b\)) are possible and dividing it by the total number of possible outcomes (all permutations of the letters).
Since we've determined there are 60 favorable outcomes when \(a\) precedes \(b\), and a total of 120 possible permutations, we use the basic probability formula:

• Probability = Number of Favorable Outcomes / Total Number of Outcomes

Thus, the probability is \(\frac{60}{120} = \frac{1}{2}\). This simple calculation demonstrates how probability theory can be applied to solve permutation problems and helps in predicting outcomes with mathematical precision.