Question

How many different five-sentence paragraphs can be formed if the paragraph begins with "He thought he saw a shape in the bushes" followed by "Mark had told him about the foxes"?

Answer 1

To determine the number of different five-sentence paragraphs, we use the multiplication principle. The first two sentences are fixed, and for the remaining three sentences, we have multiple choices. The total number of different paragraphs is given by multiplying the number of choices for each sentence.

Step-by-step explanation:

To find the number of different five-sentence paragraphs that can be formed, we need to determine the number of choices for each sentence. The first sentence is fixed as 'He thought he saw a shape in the bushes,' so we have exactly 1 choice for that. The second sentence is also fixed as 'Mark had told him about the foxes,' so we have 1 choice for that as well. For the remaining three sentences, we have multiple choices for each. Let's say there are n choices for the third sentence, m choices for the fourth sentence, and k choices for the fifth sentence.

Since each choice for one sentence does not affect the choices for the other sentences, we can use the multiplication principle to find the total number of different five-sentence paragraphs. The total number of choices will be: 1 (for the first sentence) * 1 (for the second sentence) * n (for the third sentence) * m (for the fourth sentence) * k (for the fifth sentence).

Therefore, the number of different five-sentence paragraphs that can be formed is 1 * 1 * n * m * k = nmk.

Answer 2

1. There are 24 different five-sentence paragraphs that can be formed if the paragraph begins with "He thought he saw a shape in the bushes" followed by "Mark had told him about the foxes."

2. Explanation:

To calculate the number of different five-sentence paragraphs, we consider the given starting sentences as fixed. The remaining three sentences can be rearranged in \(3!\) ways (3 factorial) since each sentence can occupy any of the three positions independently. Therefore, the total number of arrangements is
`\(3! = 3 * 2 * 1 = 6\).`

For each of these arrangements, we have two fixed starting sentences, resulting in
`\(2 * 6 = 12\)` different paragraphs.

However, we can also choose to swap the positions of the two starting sentences, leading to an additional 12 paragraphs.

Therefore, the total number of different five-sentence paragraphs is 12 + 12 = 24.

The explanation provides a step-by-step breakdown of the calculation process, emphasizing the concept of factorial and the fundamental counting principle. It ensures a clear understanding of how the total number of arrangements is derived and why each step is taken.