A boy has 5 Tarzan books, 4 Harry Potter books and 4 dictionaries he’s going to put on a shelf. How many possible combinations can he arrange the books on the shelf if no conditions are set? if the tarzan books are to be together, the harry potter books together and the dictionaries together?

To solve this problem, we'll calculate the number of ways the books can be arranged under two different scenarios:

1. **No conditions are set**: The books can be arranged in any order.
2. **Tarzan books together, Harry Potter books together, and dictionaries together**: Books of each type must be grouped together.

### 1. No Conditions Set

When there are no conditions, we treat all the books as distinct items that can be arranged in any order. The total number of books is:

\[
5 \, (\text{Tarzan}) + 4 \, (\text{Harry Potter}) + 4 \, (\text{Dictionaries}) = 13 \, \text{books}
\]

The number of possible arrangements is the factorial of the total number of books:

\[
13! = 13 \times 12 \times 11 \times \ldots \times 2 \times 1
\]

### 2. Tarzan Books Together, Harry Potter Books Together, and Dictionaries Together

If each set of books must stay together, we treat each group as a single unit. This means we have 3 groups: Tarzan, Harry Potter, and Dictionaries.

#### Step 1: Arrange the Groups

First, we arrange the 3 groups. The number of ways to arrange 3 groups is:

\[
3! = 6
\]

#### Step 2: Arrange Books Within Each Group

Next, we arrange the books within each group:

- **Tarzan books**: 5 books can be arranged in \(5!\) ways.
- **Harry Potter books**: 4 books can be arranged in \(4!\) ways.
- **Dictionaries**: 4 books can be arranged in \(4!\) ways.

#### Step 3: Calculate the Total Arrangements

To find the total number of ways to arrange the books with the given conditions, we multiply the number of ways to arrange the groups by the number of ways to arrange the books within each group:

\[
3! \times 5! \times 4! \times 4!
\]

Let's compute these factorials:

\[
3! = 6, \quad 5! = 120, \quad 4! = 24
\]

Now substitute these values into the formula:

\[
6 \times 120 \times 24 \times 24 = 414,720
\]

### Final Answers

1. **No conditions set**: The number of possible arrangements is \(13! \approx 6,227,020,800\).
2. **Books grouped together by type**: The number of possible arrangements is \(414,720\).

Would you like more details, or do you have any questions?

Here are 5 related questions to consider:
1. What if the Harry Potter books had to be in a specific order within their group?
2. How would the answer change if two books were identical?
3. What if the dictionaries must be placed at both ends of the shelf?
4. How would we calculate the combinations if we added another set of books?
5. Can you find the number of arrangements if only the Tarzan books had to be together?

**Tip:** Grouping objects and using factorials are common methods to calculate the number of arrangements in combinatorial problems.

A boy has 5 Tarzan books, 4 Harry Potter books, and 4 dictionaries he’s going to put on a shelf. How many possible combinations can he arrange the books on the shelf if no conditions are set? If the Tarzan books are to be together, the Harry Potter books together, and the dictionaries together?

Learn how to calculate the number of possible combinations for arranging 5 Tarzan books, 4 Harry Potter books, and 4 dictionaries on a shelf with or without grouping conditions. This combinatorics problem involves factorials, grouping, and permutations. Ideal for Grades 10-12.

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