1. The problem is to understand and solve questions related to permutations and combinations. 2. Permutations refer to the arrangement of objects where order matters, and combinations refer to the selection of objects where order does not matter. 3. The formula for permutations of $n$ objects taken $r$ at a time is $$P(n,r) = \frac{n!}{(n-r)!}$$ where $n!$ denotes factorial of $n$. 4. The formula for combinations of $n$ objects taken $r$ at a time is $$C(n,r) = \frac{n!}{r!(n-r)!}$$. 5. Important rules: - Factorial $n! = n \times (n-1) \times (n-2) \times \cdots \times 1$. - For permutations, order matters; for combinations, order does not matter. 6. Example: Find the number of ways to arrange 3 books out of 5 on a shelf (permutations). 7. Using the permutation formula: $$P(5,3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{120}{2} = 60$$. 8. So, there are 60 ways to arrange 3 books out of 5. 9. Example: Find the number of ways to select 3 books out of 5 (combinations). 10. Using the combination formula: $$C(5,3) = \frac{5!}{3!2!} = \frac{120}{6 \times 2} = \frac{120}{12} = 10$$. 11. So, there are 10 ways to select 3 books out of 5 without regard to order. This explanation covers the basics of permutations and combinations with formulas and examples.