Factorials are a compact way to multiply a whole number by every positive whole number below it. They appear whenever order matters, such as arranging books, assigning race finishes, or counting possible passwords. The symbol n! is read as n factorial, and it grows much faster than ordinary powers or linear expressions.
Understanding factorials gives students a strong foundation for permutations, combinations, probability, and series.
Key Facts
- Definition for positive integers: n! = n × (n - 1) × ... × 2 × 1
- Base case: 0! = 1
- Recursive rule: n! = n × (n - 1)!
- Examples: 4! = 4 × 3 × 2 × 1 = 24 and 6! = 720
- Permutations of n distinct objects: n!
- Arrangements of r objects chosen from n: P(n, r) = n! / (n - r)!
Vocabulary
- Factorial
- The product of a whole number and every positive whole number less than it.
- Permutation
- An arrangement of objects where the order of the objects matters.
- Combination
- A selection of objects where the order of the objects does not matter.
- Recursive definition
- A definition that describes a value using a smaller version of the same type of value.
- Sample space
- The set of all possible outcomes in a probability situation.
Common Mistakes to Avoid
- Writing 5! = 5 × 4 instead of 5 × 4 × 3 × 2 × 1. A factorial continues multiplying all positive integers down to 1.
- Thinking 0! = 0. Zero factorial equals 1 because it represents one way to arrange nothing and makes the recursive rule 1! = 1 × 0! work.
- Using n! when only some items are being arranged. If choosing and ordering r items from n, use P(n, r) = n! / (n - r)! instead of n!.
- Canceling factorials incorrectly, such as treating 7! / 5! as 7 / 5. Since 7! = 7 × 6 × 5!, the quotient is 7 × 6 = 42.
Practice Questions
- 1 Compute 7! and show the multiplication used.
- 2 A club has 8 members. How many ways can it choose a president, vice president, and secretary if no person can hold more than one office?
- 3 Explain why 0! = 1 makes sense using either the idea of arranging zero objects or the recursive rule for factorials.