1. **Stating the problem:** We want to understand permutations, which are arrangements of objects in a specific order. 2. **Formula for permutations:** The number of ways to arrange $n$ distinct objects is given by the factorial of $n$, written as $$n! = n \times (n-1) \times (n-2) \times \cdots \times 1.$$ For example, for 3 objects, the number of permutations is $$3! = 3 \times 2 \times 1 = 6.$$ 3. **Permutations of a subset:** If we want to arrange $r$ objects out of $n$ distinct objects, the formula is $$P(n,r) = \frac{n!}{(n-r)!}.$$ This counts the number of ordered arrangements of $r$ objects chosen from $n$. 4. **Example 1:** How many ways can we arrange 3 books out of 5 on a shelf? Using the formula: $$P(5,3) = \frac{5!}{(5-3)!} = \frac{120}{2} = 60.$$ So, there are 60 different ways. 5. **Example 2:** Arrange all 4 letters of the word "MATH". Since all letters are distinct, the number of permutations is $$4! = 24.$$ 6. **Important rules:** - Order matters in permutations. - No repetition unless stated. - Factorial grows very fast, so for large $n$, permutations become very large. 7. **Summary:** Permutations count the number of ways to order objects. Use $n!$ for all objects and $P(n,r)$ for subsets. Final answer for example 1: 60 ways. Final answer for example 2: 24 ways.