Factorial Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Factorial.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n: n! = n \cdot (n-1) \cdots 2 \cdot 1.
Factorial counts the number of ways to arrange n distinct objects in a row — for 3 items, there are 3! = 6 possible orderings.
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: Factorials grow extremely fast: 10! = 3{,}628{,}800. The key identity is n! = n \cdot (n-1)!, which defines the recursive structure.
Common stuck point: 0! = 1 by definition (or convention), not by direct multiplication — this edge case is essential for combinatorial formulas to work when choosing all or no items.
Sense of Study hint: Try writing the countdown: 5! = 5 x 4 x 3 x 2 x 1. Start from n and multiply down to 1. Remember 0! = 1 by definition.
Worked Examples
Example 1
easySolution
- 1 Recall the factorial definition: n! = n \times (n-1) \times \cdots \times 2 \times 1. Write out 7!: 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
- 2 Multiply step by step: 7 \times 6 = 42, then 42 \times 5 = 210, then 210 \times 4 = 840, then 840 \times 3 = 2520.
- 3 Complete the product: 2520 \times 2 = 5040, so 7! = 5040.
Answer
Example 2
mediumPractice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easyExample 2
mediumRelated Concepts
Background Knowledge
These ideas may be useful before you work through the harder examples.