Problem 39 - Algebra and Trigonometry

Question

In the Cambridge Encyclopedia of Language (Cambridge University Press, 1987), author David Crystal presents five sentences that make a reasonable paragraph regardless of their order. The sentences are as follows: • Mark had told him about the foxes. • John looked out the window. • Could it be a fox? • However, nobody had seen one for months. • He thought he saw a shape in the bushes. How many different five-sentence paragraphs can be formed if the paragraph begins with “He thought he saw a shape in the bushes” and ends with “John looked out of the window”?

Step-by-Step Solution

Answer

There are 6 different five-sentence paragraphs that can be formed if the paragraph begins with 'He thought he saw a shape in the bushes' and ends with 'John looked out of the window'.

Step 1: Identify the fixed positions

‘He thought he saw a shape in the bushes’ must always start and ‘John looked out of the window’ must always finish. So, these two hold fixed positions and only the remaining three sentences can be rearranged in any order.

Step 2: Calculate the Permutations

Calculate the number of ways 3 objects (the three sentences) can be arranged. That's a permutation problem and the formula to solve it is \(P(n) = n!\) where \(n\) is the number of objects and \(!\) indicates factorial. Here, \(n = 3\).

Step 3: Apply Permutation Formula

Apply the permutation formula to find the number of arrangements for the remaining three sentences. The calculation is as follows: \(P(3) = 3! = 3×2×1 = 6\).

Key Concepts

Factorial

Understanding the concept of a factorial is crucial when solving permutation problems. A factorial, denoted by an exclamation point (!), represents the product of all positive integers up to a certain number. For instance, the factorial of 3, written as 3!, is the product of all positive integers from 1 to 3, which is calculated as:
\[\begin{equation}3! = 3 \times 2 \times 1 = 6\end{equation}\]
Factorials grow rapidly with each increment of the number; for example, 4! is 24, 5! is 120, and so on. Factorials are a basic building block in combinatorics, the branch of mathematics dealing with counting and arranging objects.

Why Use Factorial in Permutation Problems?

Factorials are used in permutation problems because they provide a simple way to calculate the total number of distinct arrangements for a set of objects. In permutation problems, every minute detail matters, as the arrangement of every single element affects the outcome.

Arrangements

The concept of arrangements refers to the different ways in which a set of objects can be ordered or organized. When solving problems that ask for the total number of possible arrangements, it's essential to determine whether certain conditions must be met. In permutations, each arrangement is unique based on the order of its elements.

How Do Arrangements Apply to Our Problem?

In the Cambridge Encyclopedia exercise, we're tasked with finding the total number of arrangements for sentences in a paragraph with specified starting and ending sentences. Since two sentences are fixed, we focus on arranging the remaining three.

• The first and the last sentences act as our anchors.
• The rest of the sentences can be shuffled between these anchors.
• Arrangements of these 'middle' sentences create unique paragraphs.

So, by understanding how to apply permutations to these middle sentences, we can determine the total number of unique paragraphs that can be formed.

Order of Sentences

In textual problems like the one encountered in the Cambridge Encyclopedia, the order of sentences is significant as it can alter the meaning or coherence of a paragraph. While many permutation problems involve objects that don't have intrinsic sequencing, sentences must be arranged in an order that retains the intended narrative or logical flow.

Advanced Implications of Sentence Ordering

Order of sentences is particularly important in literature, legal documentation, and coding of information. In our exercise, by fixing the first and last sentences, we are determining the 'frame' of the paragraph, ensuring there is a discernible start and end. The arrangement of the remaining sentences then dictates the progression of thoughts or events within that frame. Understanding the effect of sentence order helps to comprehend the broader implications of arrangements in language and communication structures.