1. **State the problem:** We want to find the number of ways to arrange 4 different books on a shelf. 2. **Formula used:** The number of ways to arrange $n$ different items in order is given by the factorial of $n$, denoted as $n!$. 3. **Explanation:** Factorial means multiplying all positive integers from 1 up to $n$. For example, $4! = 4 \times 3 \times 2 \times 1$. 4. **Calculate:** $$ 4! = 4 \times 3 \times 2 \times 1 = 24 $$ 5. **Answer:** There are 24 different ways to arrange 4 different books on a shelf.