In how many ways can 12 books be arranged on a shelf if 4 particular books must always be together.

Answer

To solve the problem of arranging 12 books on a shelf where 4 particular books must always be together, we can follow these steps:

### Step 1: Treat the 4 particular books as a single unit
Since the 4 particular books must always be together, we can consider them as one single unit or "block". This means instead of having 12 individual books, we now have 9 units to arrange: the block of 4 books and the other 8 individual books.

### Step 2: Count the total units
Now we have:
- 1 block of 4 books
- 8 other individual books

This gives us a total of \( 9 \) units to arrange.

### Step 3: Calculate the arrangements of the 9 units
The number of ways to arrange \( n \) distinct units is given by \( n! \). Therefore, the number of ways to arrange our 9 units is:
\[ 9! = 362880 \]

### Step 4: Arrange the books within the block
Within the block of 4 books, these books can also be arranged among themselves. The number of ways to arrange 4 books is given by \( 4! \):
\[ 4! = 24 \]

### Step 5: Calculate the total arrangements
To find the total number of arrangements of the books on the shelf, we multiply the arrangements of the 9 units by the arrangements of the 4 books within the block:
\[ \text{Total arrangements} = 9! \times 4! = 362880 \times 24 = 8709120 \]

Thus, the total number of ways to arrange the 12 books on the shelf, with the condition that the 4 particular books are always together, is **8,709,120**.