Algebra worksheet - Free Printable

Problem Statement:

The problem asks us to determine the number of ways to arrange 5 distinct books on a shelf such that two specific books, say Book A and Book B, are always next to each other.

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Solution:

#### Step 1: Treat Book A and Book B as a single "block"
Since Book A and Book B must always be next to each other, we can treat them as a single unit or "block." This reduces the problem from arranging 5 distinct books to arranging 4 units:
- The "AB" block (or "BA" block, since they can switch positions within the block).
- The remaining 3 individual books.

Thus, we now have 4 units to arrange: the "AB" block and the 3 other books.

#### Step 2: Arrange the 4 units
The number of ways to arrange 4 distinct units is given by the factorial of 4:
\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]

#### Step 3: Account for the internal arrangement of the "AB" block
Within the "AB" block, Book A and Book B can switch places. There are 2 possible arrangements for the block:
- AB
- BA

Thus, for each arrangement of the 4 units, there are 2 ways to arrange Book A and Book B within their block.

#### Step 4: Combine the results
To find the total number of arrangements, we multiply the number of ways to arrange the 4 units by the number of ways to arrange Book A and Book B within their block:
\[
4! \times 2 = 24 \times 2 = 48
\]

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Final Answer:

The total number of ways to arrange the 5 books such that Book A and Book B are always next to each other is:
\[
\boxed{48}
\]