How many ways can we arrange five of the seven Harry Potter books on a shelf if Harry Potter and the Chamber of Secrets must be one of them?

To solve this problem, we need to determine how many ways we can arrange five out of the seven Harry Potter books on a shelf, with the condition that **Harry Potter and the Chamber of Secrets (Book 2)** must be included.

### Step 1: Fix Harry Potter and the Chamber of Secrets
Since "Chamber of Secrets" must be one of the five books, we can treat it as a fixed book in our selection.

### Step 2: Choose 4 more books from the remaining 6
After selecting the "Chamber of Secrets," we need to choose 4 additional books from the remaining 6 books. The number of ways to choose 4 books from 6 is given by the combination formula:
\[
\binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6 \times 5}{2 \times 1} = 15
\]
So, there are 15 ways to choose 4 more books from the remaining 6.

### Step 3: Arrange the 5 books
Once we have selected 5 books (including "Chamber of Secrets"), the number of ways to arrange these 5 books on a shelf is \(5!\):
\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

### Step 4: Calculate the total number of arrangements
Now, multiply the number of ways to choose the 4 additional books by the number of ways to arrange the 5 books:
\[
15 \times 120 = 1800
\]

### Final Answer:
There are **1,800** ways to arrange five of the seven Harry Potter books on a shelf if "Harry Potter and the Chamber of Secrets" must be one of them.

Do you have any questions or would you like further details?

Here are five related questions to consider:
1. How would the result change if "Chamber of Secrets" was optional?
2. How many ways are there to arrange all seven books on the shelf?
3. What if two specific books (e.g., "Chamber of Secrets" and "Prisoner of Azkaban") must be included in the arrangement?
4. How many ways can we arrange five books if no restrictions are placed on which ones must be included?
5. How does the number of arrangements change if the order on the shelf doesn’t matter?

**Tip:** For problems involving "choosing" and "arranging," think about combinations when order doesn’t matter and permutations when order does!

Discover how to calculate the number of ways to arrange 5 out of 7 Harry Potter books on a shelf, with the condition that Harry Potter and the Chamber of Secrets is included. This combinatorics problem is solved using combination and permutation formulas. Suitable for students in Grades 9-12.

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