Factorial Formula
The Formula
When to use: Factorial counts the number of ways to arrange n distinct objects in a row — for 3 items, there are 3! = 6 possible orderings.
Quick Example
Notation
What This Formula Means
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.
Formal View
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
mediumCommon Mistakes
- Thinking 0! = 0 — by definition, 0! = 1
- Computing n! + m! as (n + m)! — factorials do not distribute over addition
- Underestimating how fast factorials grow — 20! is over 2 \times 10^{18}, far too large for casual computation
Why This Formula Matters
Factorials appear in permutations, combinations, the binomial theorem, Taylor series, and probability — they are the building block of discrete counting.
Frequently Asked Questions
What is the Factorial formula?
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.
How do you use the Factorial formula?
Factorial counts the number of ways to arrange n distinct objects in a row — for 3 items, there are 3! = 6 possible orderings.
What do the symbols mean in the Factorial formula?
n! reads 'n factorial'; by convention 0! = 1
Why is the Factorial formula important in Math?
Factorials appear in permutations, combinations, the binomial theorem, Taylor series, and probability — they are the building block of discrete counting.
What do students get wrong about Factorial?
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.
What should I learn before the Factorial formula?
Before studying the Factorial formula, you should understand: multiplication.