luna has 4 different math books and 5 different physics books, how many ways can she arrange the books on a shelf given that the physics books must be together as a block

To solve this, we can break the problem into two steps:

### Step 1: Treat the 5 physics books as a single block.
Since the physics books must be together, we can consider them as one "super book." This simplifies the problem into arranging 4 math books and 1 "super physics book," meaning we now have 5 items to arrange.

The number of ways to arrange 5 items (4 math books and 1 block of physics books) is:

\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

### Step 2: Arrange the 5 physics books within the block.
Even though the 5 physics books are treated as a single block, they can still be arranged in different ways among themselves. The number of ways to arrange 5 physics books is:

\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

### Step 3: Multiply the results of Step 1 and Step 2.
To find the total number of ways to arrange the books, multiply the number of ways to arrange the 5 items (Step 1) by the number of ways to arrange the physics books within the block (Step 2):

\[
120 \times 120 = 14,400
\]

Thus, Luna can arrange the books in **14,400** different ways.

Would you like more details or have any questions? Here are some related questions:

1. What if the math books must also stay together as a block?
2. What if Luna wants to arrange only 3 math books?
3. How does the answer change if two of the physics books are identical?
4. How would the problem change if Luna had 6 different physics books?
5. What if the books can be arranged freely without any blocks?

**Tip:** When books are grouped in blocks, treat the block as a single item to simplify the calculation.

Luna has 4 different math books and 5 different physics books, how many ways can she arrange the books on a shelf given that the physics books must be together as a block?

Learn how to calculate the number of ways to arrange 4 math books and 5 physics books when the physics books must stay together as one block. This combinatorics problem involves factorials and the Fundamental Principle of Counting. Suitable for students in Grades 9-12.

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